Abstract Mathematical Question
Eldon Moritz
mathematical results" thread, in this newsgroup.
Our question:
Two coins were flipped and at least one is a tail. What are the
chances for two tails? or
Two coins were flipped and at least one is a head. What are the
chances for two heads?
We seem to have agreed that the two are actually the same question,
and should have the same answer. (we haven't agreed upon much else)
I introduced the following two math models.
Call this Math Model 1.
Two coins were flipped.
1) P (2 heads | at least one head)
2) = P (2 heads AND at least one head) / P (at least one head)
3) = P (2 heads) / P (at least one head)
4) = (1/4) / (1/4+1/4+1/4 +0)
5) = 1/3
Math Model 2
Two coins were flipped
1) P (2 heads | Statement Qe)
2) = P (2 heads AND Statement Qe) / P (Statement Qe)
3) = P (2 heads) / P (Statement Qe)
4) = (1/4) / (1/4 + (1/2*1/4)+(1/2*1/4)+0)
5) = 1/2
Qe can be either "Two coins were flipped and at least one is a head"
or "Two coins were flipped and at least one is a tail." Either
statement would get correct answer 1/2, using Model 2.
Model 1 and Model 2 are different models with different answers.
There is a coin flip sequence for Model 1, and a coin flip sequence
for Model 2. They are different coin flip sequences. In Model 1, there
is more known information.
There is a statement, and question which would give the exact
hypothesis for Model 1, and correctly defines the corresponding coin
flip sequence. Call it Qg. It has correct answer 1/3.
There is a statement, and question which gives the exact hypothesis
for Model 2, and correctly defines the corresponding coin flip
sequence. Call it Qe. It has correct answer 1/2.
I have stated that model 1 seems to be the correct model for the
probability for two, given at least one. It gets correct answer 1/3
for that statement, and as such, defines what is meant by the term
"given at least one." It does not define what is meant by "at least
one is."
Qe has correct answer 1/2.
There is a working model for Qe, whereas the exact wording is, "Two
coins were flipped and at least one is a head. What are the chances
for two heads?" Dr. Holt, for one, has agreed that this is true. He
has agreed in the aforementioned thread. Qe has minimum information.
There isn't enough information in Qe to use model 1.
Dr. Holt holds out for the possibility of another interpretation of
the question. In other words, he says that the question may be
ambiguous. So far, he, nor anyone else has come forth with a correct
working model for another interpretation.
Qe, and Qg have different answers. They define different coin flip
sequences. If they are the exact same question, then we have
ambiguity.
The esteemed Dr. David Ullrich, has vehemently denied Moritz'
solution. He has refused to discuss the solution, but has claimed
higher authority.
Ullrich introduced Question Q. We argued Q earlier in this same
newsgroup.
Q: A woman has two children, at least one of which is a boy. What is
the probability that she has two boys?
That's okay with me. I consider Q, and Qe, to be mathematical
equivalents. Q was introduced to this this newsgroup after being asked
in Parade magazine by Marilyn vos Savant. This is a family magazine
with national distribution to ordinary people. It comes in the
newspaper on Sunday mornings.
Then I offered the following, on April 4 in the "Counter-intuitive
mathematical results" thread:
In my arguments, we know how we obtained this information. We obtained
our information from Q. BB happened and Q was said, or, BG happened
and Q was said, or, GB happened and Q was said. Are the three still
equally likely?
To this, the Illustrious professor, Dr. Ullrich replied:
<start quote>What an extremely dumb fuck you are. I mean whether you
agree with me or not, it's _really_ fucking dumb not to know by now
what my answer is: None of those three have anything to do with the
answer to Q. Q is an abstract mathematical question, has nothing to do
with what happened or who said what - we are _given_ certain
information and asked about a certain probability given that
information. <end of quote>
My question for this forum:
What is an abstract mathematical question, and how do we know when we
have one? Our question appeared in a family magazine. How were we to
know?
Methinks the perfesser thought that shit up to protect his ignorant
position.
Eldon:)
Arthur
"Eldon Moritz" <elmo...@yahoo.com> wrote:
> We have been discussing coin flips in the "Counter-intuitive
> mathematical results" thread, in this newsgroup.
>
> Our question:
> Two coins were flipped and at least one is a tail. What are the
> chances for two tails? or
> Two coins were flipped and at least one is a head. What are the
> chances for two heads?
>
> We seem to have agreed that the two are actually the same question,
> and should have the same answer. (we haven't agreed upon much else)
right... I've seen many arguments on this one before. IMO, the solution is
extremely simple. Just write out the solution space:
hh, ht, th, tt, all 4 with equal probability 1/4. Given the info that at
least one is head (h), the solution space gets reduced to:
hh, ht, th, all with probability 1/3. So P(hh) = 1/3.
This is equivalent to model 1. I don't see that quickly what you mean by
model 2...
- Arthur
Doug Magnoli
> "Eldon Moritz" <elmo...@yahoo.com> wrote:
> > We have been discussing coin flips in the "Counter-intuitive
> > mathematical results" thread, in this newsgroup.
> >
> > Our question:
> > Two coins were flipped and at least one is a tail. What are the
> > chances for two tails? or
> > Two coins were flipped and at least one is a head. What are the
> > chances for two heads?
> >
> > We seem to have agreed that the two are actually the same question,
> > and should have the same answer. (we haven't agreed upon much else)
>
> right... I've seen many arguments on this one before. IMO, the solution is
> extremely simple. Just write out the solution space:
> hh, ht, th, tt, all 4 with equal probability 1/4. Given the info that at
> least one is head (h), the solution space gets reduced to:
> hh, ht, th, all with probability 1/3. So P(hh) = 1/3.
> This is equivalent to model 1. I don't see that quickly what you mean by
> model 2...
>
> - Arthur
I think model 2 corresponds to the situations where we have flipped two
coins, and I pull one of them out of the hat and say, 'and _this_ one came up
heads.' Now what's the probability of two heads?
It's clearly different information from that in the problem you just
amplified.
So it comes down to is what does it mean to say "one of them came up heads."
If it means some one of them, and we don't know which, the problem means one
thing. If it means *this* one came up heads, it means something else.
-Doug Magnoli
[Remove the two and the three for email.]
David C. Ullrich
>We have been discussing coin flips in the "Counter-intuitive
>mathematical results" thread, in this newsgroup.
>
>Our question:
>Two coins were flipped and at least one is a tail. What are the
>chances for two tails? or
>Two coins were flipped and at least one is a head. What are the
>chances for two heads?
>
>We seem to have agreed that the two are actually the same question,
>and should have the same answer. (we haven't agreed upon much else)
[...]
>
>I have stated that model 1 seems to be the correct model for the
>probability for two, given at least one. It gets correct answer 1/3
>for that statement, and as such, defines what is meant by the term
>"given at least one." It does not define what is meant by "at least
>one is."
In case anyone can't figure out what's meant here: He's insisting
that the two questions
"Two coins were flipped and at least one is a tail. What are the
chances for two tails?"
and
"Two coins were flipped. Given that at least one is a tail, what
are the chances for two tails?"
are different.
Honest. I point this out expliictly for the benefit of readers
who might try to figure out what his point is, and might
have a hard time, because they'd never imagine that
someone who spoke fluent English would think that
the two were different questions.
If anyone wonders why I tend to call the guy a dumb fuck, you
should understand that it's not intended as an insult. Eldon
says he calls people dumb fucks in order to get their attention
(even in contexts where they were already paying attention.)
Now, when I call him a dumb fucking asshole, _that's_
intended as an insult. Has to do with various things, one
being the fact that he continues to insist that I haven't
addressed his arguments, even though I have.
It's very curious. He keeps asking me to explain what's
wrong with what he says, but when I do he never seems
to realize that I'm simply doing so, instead we get stuff
like "Methinks the perfesser thought that shit up to protect
his ignorant position." Hence the phrase "dumb fucking
asshole".
>Eldon:)
******************
David C. Ullrich
Eldon Moritz
********************
> > My question for this forum:
> > What is an abstract mathematical question, and how do we know when we
> > have one? Our question appeared in a family magazine. How were we to
> > know?
> >
********************
You ignored my question.
> > Methinks the perfesser thought that shit up to protect his ignorant
> > position.
> >
> > Eldon:)
Thanks Arthur,
You have re-iterated what I call the conventional one thirder
argument. We started with four, reduced to three. My argument says
that there isn't enough information in the problem statement to now
know that we have three equally likely. There is no evidence in the
statement that the writer of the question knew the outcome of both
coins.
My question for this thread, however was, and is.
What is an abstract mathematical question?
Eldon
Eldon Moritz
The other is a mathematical definition.
That's different.
I say, and I'll say again. "Two coins were flipped and at least one is
a tail. What are the chances for two tails?" Has correct answer 1/2.
"Two coins were flipped. Given that at least one is a tail, what are
the chances for two tails?" Seems to be 1/3, by mathematical
definition. The two questions are the same, or different, depending
upon how you define "given at least one."
Manipulate the numbers in math model 1 and re define "Given at least
one". This would not affect the other question.
Whoooosh......straight over you head
I asked what is meant by given at least one? You wouldn't say.
I said, "If a student asked, What is meant by given at least one, what
would you tell him/her?" You wouldn't say. Given at least one defines
a hypothesis. What is it? Two coins flip four ways. To get correct
answer 1/3, the flip has to be manipulated. You can't flip two coins
three ways without manipulating the flip. How did you do this?
Whooooosh....straight over your head.
You said my argument was wrong because this is a complex mathematical
problem. I asked what that is. You won't say.
> It's very curious. He keeps asking me to explain what's
> wrong with what he says, but when I do he never seems
> to realize that I'm simply doing so, instead we get stuff
> like "Methinks the perfesser thought that shit up to protect
> his ignorant position." Hence the phrase "dumb fucking
> asshole".
>
I have shown you a working model for my 'interpretation'. It works.
I said, "Show me a working model which works for your argument, and
I'll capitulate." So far, you haven't done so. That seems to go
Whoooooosh.. Straight over your head.
You have incorrectly paraphrased my argument several times. I don't
think you understand it. Whoooosh...
I think, and I might be wrong, that the illustrious professor said
"fuck" in sci.math before I did.
I said something about what the asker of our question had to know. You
said "there was no asker." That's wrong. So far, you haven't
capitulated.We have a written question. It was written by someone, or
something.. Whatever asked the question, call whatever the asker.
Whoooosh.....
I'm trying to tell you something which you don't seem to be willing to
grasp.
I'm the asshole?
Eldon
Virgil
<349f5619.03040...@posting.google.com>,
elmo...@yahoo.com (Eldon Moritz) wrote:
>
> Whoooosh......straight over you head
>
> Whooooosh....straight over your head.
>
> Whoooooosh.. Straight over your head.
>
> Whoooosh...
>
> Whoooosh.....
>
> I'm the asshole?
>
> Eldon
The way you are whooshing all over the place, one does not
really want to say in public what you are.
Randy Poe
This is the subject of another endless thread. Eldon Moritz
believes that the problem statement:
"Two coins are flipped. Given at least one is a head..."
is different from
"Two coins are flipped. At least one is a head..."
In the second case, he believes the problem is saying that
we lack the omniscience of the problem-stater. Instead, there
is some person who is observing the flips and reports
that there is a head. This person might sometimes say
"there is a tail" in HT or TH cases.
- Randy
Gus Gassmann
[...lots of stuff omitted...]
> I think, and I might be wrong, that the illustrious professor said
> "fuck" in sci.math before I did.
As a quick search in google will tell, you used "dumb fuck", specifically
directed at him, on March 31st. It took two days before he retaliated.
So you were wrong on this one.
> I said something about what the asker of our question had to know. You
> said "there was no asker." That's wrong. So far, you haven't
> capitulated.We have a written question. It was written by someone, or
> something.. Whatever asked the question, call whatever the asker.
>
> Whoooosh.....
>
> I'm trying to tell you something which you don't seem to be willing to
> grasp.
>
> I'm the asshole?
This one, I think you nailed, though...
Randy Poe
> I said something about what the asker of our question had to know. You
> said "there was no asker." That's wrong. So far, you haven't
> capitulated.We have a written question. It was written by someone, or
> something.. Whatever asked the question, call whatever the asker.
You have distorted, either deliberately or by misunderstanding,
the discussion of the "asker".
Of course somebody wrote the problem. That means there is
an author asking YOU, the reader something.
The point of contention is whether a statement like
"One of the coins is a heads" implies that the question
author had to obtain this information by asking
a question of SOMEBODY ELSE. Somebody who, in your
interpretation, might have answered differently even
if there was in fact a heads among the flips.
For some reason you accept that the question author
can have test-givers omniscience when saying "given
that one of the coins is a heads" but when the
author leaves out the word "given" he loses his
omniscience and must rely on polls and unreliable
third parties.
Suppose I ask you another old chestnut: "On a mysterious
island you meet three natives. One always tells the
truth. One always lies. One alternately lies and
tells the truth...."
The word "given" is absent. Do you then assume that
in fact my information about truth-telling and lie-telling
is unreliable?
"A hen-and-a-half lays an egg-and-a-half in a day-and-a-half.
How many days does it take 10 hens to lay 10 eggs?"
Again, the word "given" is absent. Thus, you feel you can't
take the first sentence as true for purposes of solving
the problem?
Closer to home (I just made this up): There are 50 children
is Mrs. Smith's homeroom class. 15 have peanut-butter
sandwiches in their lunch today. 10 have apples...
Do you assume that I can only have gleaned this information
by waiting for the children to volunteer the info? That
there might be more than 15 with peanut-butter sandwiches,
but only those 15 said something? That because I didn't
include the word "given", I didn't know it for sure when
I composed the problem about this fictitious class and
I am allowing this description to include classes where
the number is different from 15?
- Randy
David C. Ullrich
I think that given that you're replying to a rhetorical question
your answer is a little ambiguous. Or might be misinterpreted
by a sufficiently dumb fuck, in any case.
You meant "yes", right?
******************
David C. Ullrich
Eldon Moritz
> Eldon Moritz wrote:
>
> [...lots of stuff omitted...]
>
> > I think, and I might be wrong, that the illustrious professor said
> > "fuck" in sci.math before I did.
>
> As a quick search in google will tell, you used "dumb fuck", specifically
> directed at him, on March 31st. It took two days before he retaliated.
This happened, and I remember it well. It is in the "Counter-intuitive
mathematical results" thread. A further, quick search in google will
find many posts by the illustrious professor in which he said fuck;
some of them as far back as 2001, and maybe earlier.
> So you were wrong on this one.
>
Not if you read the statement precisely. I still think, and I still
may be wrong that the March 31, 2003 post was the first in which I
said fuck in sci.math.
Your apology is accepted.
> > I said something about what the asker of our question had to know. You
> > said "there was no asker." That's wrong. So far, you haven't
> > capitulated.We have a written question. It was written by someone, or
> > something.. Whatever asked the question, call whatever the asker.
> >
> > Whoooosh.....
> >
> > I'm trying to tell you something which you don't seem to be willing to
> > grasp.
> >
> > I'm the asshole?
>
> This one, I think you nailed, though...
No one, not even me, can be wrong every time.
Eldon
Eldon Moritz
>
> >We have been discussing coin flips in the "Counter-intuitive
> >mathematical results" thread, in this newsgroup.
> >
> >Our question:
> >Two coins were flipped and at least one is a tail. What are the
> >chances for two tails? or
> >Two coins were flipped and at least one is a head. What are the
> >chances for two heads?
> >
> >We seem to have agreed that the two are actually the same question,
> >and should have the same answer. (we haven't agreed upon much else)
> [...]
> >
> >I have stated that model 1 seems to be the correct model for the
> >probability for two, given at least one. It gets correct answer 1/3
> >for that statement, and as such, defines what is meant by the term
> >"given at least one." It does not define what is meant by "at least
> >one is."
>
> In case anyone can't figure out what's meant here: He's insisting
> that the two questions
>
> "Two coins were flipped and at least one is a tail. What are the
> chances for two tails?"
>
Call this B'.
> and
>
> "Two coins were flipped. Given that at least one is a tail, what
> are the chances for two tails?"
>
Call this A'.
> are different.
>
> Honest. I point this out expliictly for the benefit of readers
> who might try to figure out what his point is, and might
> have a hard time, because they'd never imagine that
> someone who spoke fluent English would think that
> the two were different questions.
>
So, are A' and B' the same, or, are they different? If they are the
same, how could an English speaking person think they were different?
What is an abstract mathematical question? That was sprung on me, and
no one seems to have answered.
I will accept that A' is an "abstract mathematical question". I've
been calling it a "mathematical definition". I called it that because
it has the word "given". _Given_ is a special mathematical word. Use
it in math model 1 to get correct answer 1/3; then we have a
definition of what it means to say "given at least one."
The math model works. The answer is true by definition. The definition
defines a hypothesis. There is a coin flip sequence which will exactly
fullfill this hypothesis. There is a statement which will exactly
convey this hypothis, will exactly describe this coin flip sequence.
Call it Qg. It is the correct question for A'. Describe this coin flip
sequence, describe this hypothesis, then you have the answer to a
student who asks, "What is meant by the term 'given at least one'".
Whether Qg actually exists, or not, is moot, as A' is an abstract
mathematical question. The numbers work. Knowing what the exact
question is, the exact wording for Qg, is not important, unless!!!!!
!!!!Unless what!!!!! Unless, you, or a student gets it mixed up with a
similar question which does exist, and has a different answer.
Use the word _given_ in a family magazine; there may be cause for
ambiguity. I saw this question -- B' -- in Parade magazine. If the
word _given_ had been in the question, I wouldn't have gotten
involved.
So: You told me we have an abstract math question.
Yes we do:
A' is an abstract mathematical question.
B' is not.
B' is a stated question.
That's different.
Don't confuse the two, or you will never get the correct answer to B'.
So now, Professor, You can, or you can't understand how I can say they
are different questions? You will, or you won't?
Eldon Moritz
<snip>
David C. Ullrich
They're the same.
>If they are the
>same, how could an English speaking person think they were different?
This is a mystery to most of us. Could be the guy was an astonishingly
dumb fuck, or it could be that somehow he acquired a serious
misunderstanding of how the language is used in this sort of context.
Or it could be a result of his obsession with the fact that He is
Right about this and everyone Else is Wrong, leading him to
misread simple English that he'd have no trouble reading correctly
in a context unrelated to his obsession.
All sorts of possibilities. But the fact that one English-speaking
person thinks that A' and B' are different does not prove that
they actually are.
******************
David C. Ullrich
Denis Feldmann
> Gus Gassmann <hgas...@mgmt.dal.ca> wrote in message
> news:<3E91B8A8...@mgmt.dal.ca>...
>> Eldon Moritz wrote:
>>
>> [...lots of stuff omitted...]
>>
>>> I think, and I might be wrong, that the illustrious professor said
>>> "fuck" in sci.math before I did.
>>
>> As a quick search in google will tell, you used "dumb fuck",
>> specifically directed at him, on March 31st. It took two days before
>> he retaliated.
>
> This happened, and I remember it well. It is in the "Counter-intuitive
> mathematical results" thread. A further, quick search in google will
> find many posts by the illustrious professor in which he said fuck;
> some of them as far back as 2001, and maybe earlier.
>
>> So you were wrong on this one.
>>
> Not if you read the statement precisely. I still think, and I still
> may be wrong that the March 31, 2003 post was the first in which I
> said fuck in sci.math.
>
> Your apology is accepted.
This is typical of your very peculiar way to read statements. You are,
indeed, a dumb fuck. Also a extremely nasty bastard. Most of all, you are
completely uninteresting. Enter killfile mode.
Joona I Palaste
> David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<hj709vk79slr5tbv2...@4ax.com>...
[David is talking about Eldon.]
>> In case anyone can't figure out what's meant here: He's insisting
>> that the two questions
>>
>> "Two coins were flipped and at least one is a tail. What are the
>> chances for two tails?"
>>
>> and
>>
>> "Two coins were flipped. Given that at least one is a tail, what
>> are the chances for two tails?"
>>
>> are different.
>>
>> Honest. I point this out expliictly for the benefit of readers
>> who might try to figure out what his point is, and might
>> have a hard time, because they'd never imagine that
>> someone who spoke fluent English would think that
>> the two were different questions.
>>
> One is a stated question. The statement defines a coin flip sequence.
> The other is a mathematical definition.
> That's different.
No. I'm sorry, but David is clearly right here and you are wrong.
The presence or absence of the word "given" makes about as much
difference as what colour of ink you use to print the question.
> I say, and I'll say again. "Two coins were flipped and at least one is
> a tail. What are the chances for two tails?" Has correct answer 1/2.
On what planet? Flipping two coins can give the following results:
HH, HT, TH, TT. We eliminate HH because it doesn't have at least one
tail. The way we humans count, this leaves the probability of TT to
be 1/3.
> "Two coins were flipped. Given that at least one is a tail, what are
> the chances for two tails?" Seems to be 1/3, by mathematical
> definition. The two questions are the same, or different, depending
> upon how you define "given at least one."
This also has the answer 1/3. The exact same reasoning as above applies
here.
> Manipulate the numbers in math model 1 and re define "Given at least
> one". This would not affect the other question.
> Whoooosh......straight over you head
Are you talking to yourself again?
(snip completely pointless drivel about whooshing)
--
/-- Joona Palaste (pal...@cc.helsinki.fi) ---------------------------\
| Kingpriest of "The Flying Lemon Tree" G++ FR FW+ M- #108 D+ ADA N+++|
| http://www.helsinki.fi/~palaste W++ B OP+ |
\----------------------------------------- Finland rules! ------------/
"A bee could, in effect, gather its junk. Llamas (no poor quadripeds) tune
and vow excitedly zooming."
- JIPsoft
Eldon Moritz
thoroughly how A' and B' are different. You completely ignored it.
Then you claim victory. This is one more case of you not understanding
my argument. I'm an asshole when I point out that I can't tell if you
can't, or won't.
Arguing with you is like arguing with Sadaam Hussein. Note that I am
not saying that you are like Sadaam Hussein. He is an internationally
known scoundrel. You are an internationally known, well respected
educator from the great state of Oklahoma.
You argue like Hussein. No matter what the facts are, you claim
victory. You ignore my argument, then claim victory. Sadaam seems to
have taken that same tactic to the ultimate extreme. Keep ignoring my
argument, you never will understand it.
You have continually paraphrased my argument wrong. I think you don't
understand it. If you understood my argument, you would paraphrase it
right, then disagree.
You can put your argument into logical terms. I can tell you where the
false statements are.
I'll put my argument into logical terms. You can't tell me where the
false statements are, there aren't any, the truth is on my side.
My argument is for B'.
My argument gets correct answer 1/2.
I don't argue A'. I can see where you get 1/3 for A', depending upon
how you define _given at least one_. I don't argue with that.
B' gets correct answer 1/2. IF A' gets 1/2, they're the same.
If A' gets 1/3, they're different. That isn't too hard to understand.
Go point to point with me and you can't win. The truth is on my side.
Start with getting my argument right.
We have 'a' statement. That's all we have.
Because of our statement we know that there is 'a' woman. All we know
of 'the' woman is what the statement told us. What was written into
the statement.
Our statement was written about the woman. (I never did say that she
said anything)
I assume that boys and girls are equally likely.
I assume that the problem statement is true.
Those assumptions are stated on my website. They may not have been
stated in this thread. The dissenters don't seem to be dissenting
because of those assumptions.
Bill and Joe always bet($2.00 bets). When a woman has two children,
Bill always bets for two of a kind. Joe always bets for one of each.
That's a fifty fifty.
All we have to do, in order to figure the odds for their bets is
correctly answer the question.
The statement is written, "A woman has two children and at least one
is a boy. What are her chances for two boys?"
Bill and Joe already have one bet. Joe thinks to himself, "Now I have
more information. My odds have increased." He wants to double the bet.
Bill said, "Okay, but give me three to two." Now, on this woman, they
have two bets. One for $2.00 even up, and one for three to two.
Who took advantage of whom?
Did the statement writer give Joe additional information? Did he not?
If so, then Joe took advantage of Bill. If not? Then Bill took
advantage of Joe.
Compare it with the three prisoners. The prisoner thought the warden
had given him additional information. In actuality, he hadn't.
Argue with the argument, or ignore the argument. If you don't
understand it, just bad mouth it. If you understand it, and can shoot
it down. Shoot at individual points. If the argument is false, at
least one of the points is false.
Don't claim victory without shooting down at least one point.
Eldon
>
>
> ******************
>
> David C. Ullrich
Randy Poe
> > a tail. What are the chances for two tails?" Has correct answer 1/2.
>
> On what planet?
On Eldon's planet (EP), where the problem poser won't tell you
"at least one is a tail" for all instances in which there is both
a head and a tail, but only for half of them.
> Flipping two coins can give the following results:
> HH, HT, TH, TT. We eliminate HH because it doesn't have at least one
> tail. The way we humans count, this leaves the probability of TT to
> be 1/3.
But not on EP. You're assuming honesty on the part of the
test-giver, and assuming that "at least one is a tail" means
the population you are drawing from is all cases where there
is a tail present. On EP, the test-giver means for you to
understand the implied words "I WOULD TELL YOU IF I FELT
LIKE IT THAT at least one is a tail" and that's not the
entire population of flips containing one tail.
- Randy
David C. Ullrich
No, you explained why you _think_ that A' and B' are different.
>You completely ignored it.
I've explained this before. It's not _possible_ to _prove_ that A' and
B' are different, just as it's not possible to prove that they are the
same. This is because they are expressed in English. The meaning
of a string of English words is not something that can be worked
out mathematically - it's a matter of convention. If you think that
A' and B' mean different things it simply follows that you don't
understand how the language is used. I ignore the details of
your analysis of various "models" because they're simply
irrelevant, being based on a misinterpretation of the statement.
Such things _are_ defined by concensus. And I have not
seen one person agree that A' and B' are different -
everyone who's commented agrees they mean the same
thing. Many people think it's very strange that anyone
would think they're not the same.
(Um, if we know the meaning of a _clause_ P and a clause
Q we can define the meaning of "P and Q" mathematicially.
But points like whether the absence of the word "given" in
B' implies that it means something different from what A'
means is not the sort of thing that's suscpetible to proof
and disproof.)
>Then you claim victory. This is one more case of you not understanding
>my argument. I'm an asshole when I point out that I can't tell if you
>can't, or won't.
>
>Arguing with you is like arguing with Sadaam Hussein. Note that I am
>not saying that you are like Sadaam Hussein. He is an internationally
>known scoundrel. You are an internationally known, well respected
>educator from the great state of Oklahoma.
>
>You argue like Hussein. No matter what the facts are, you claim
>victory. You ignore my argument, then claim victory. Sadaam seems to
>have taken that same tactic to the ultimate extreme. Keep ignoring my
>argument, you never will understand it.
>
>You have continually paraphrased my argument wrong. I think you don't
>understand it. If you understood my argument, you would paraphrase it
>right, then disagree.
>
>You can put your argument into logical terms. I can tell you where the
>false statements are.
>
>I'll put my argument into logical terms. You can't tell me where the
>false statements are, there aren't any, the truth is on my side.
It's not that simple. At the _start_ of what you propose there's a
step where the English words are _translated_ into "logical
terms". And you're simply doing the translation incorrectly.
[...]
>
>Don't claim victory without shooting down at least one point.
You keep saying this. I _have_ shot down one point. Not
with a mathematical proof, because the point in question
is not the sort of thing that's susceptible to mathematical
proof. But the point which has been shot down is the
"point" that "at least one is a head" means something
different from "given that at least one is a head".
It's true that I haven't shot this point down by anything
more compelling than just asserting that it's not so.
But that's all that's available, because of the nature of
the point in question. If "everyone" agrees that
"... At least one is a head" means the same thing
(in A' and B') as "... Given that at least one is a head"
then they _do_ mean the same thing, because that's
how language works. And in fact everyone _does_
agree they mean the same thing.
("Everyone"? Well, I've talked to many many people
about math for many years, and that's the way
everyone I've ever spoken to uses the language.
And there has been nobody here agreeing that
they mean something different. On _this_ point,
if you want to prove you're right, by _definition_
you need to find _many_ people who _agree_
you're right. You haven't done that, not even close.)
******************
David C. Ullrich
Phil Carmody
> Joona I Palaste <pal...@cc.helsinki.fi> wrote in message news:<b6umlo$nbc$1...@oravannahka.helsinki.fi>...
>> Eldon Moritz <elmo...@yahoo.com> scribbled the following:
>> > I say, and I'll say again. "Two coins were flipped and at least one is
>> > a tail. What are the chances for two tails?" Has correct answer 1/2.
>>
>> On what planet?
>
> On Eldon's planet (EP), where the problem poser won't tell you
> "at least one is a tail" for all instances in which there is both
> a head and a tail, but only for half of them.
Eldon's _special_ planet (ESP) surely.
Note - don't try to read his mind, for obvious reasons...
Phil
(not the completely fuckwitted one)
Eldon Moritz
> > David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<hj709vk79slr5tbv2...@4ax.com>...
>
> [David is talking about Eldon.]
>
> >> In case anyone can't figure out what's meant here: He's insisting
> >> that the two questions
> >>
> >> "Two coins were flipped and at least one is a tail. What are the
> >> chances for two tails?"
> >>
> >> and
> >>
> >> "Two coins were flipped. Given that at least one is a tail, what
> >> are the chances for two tails?"
> >>
> >> are different.
> >>
> >> Honest. I point this out expliictly for the benefit of readers
> >> who might try to figure out what his point is, and might
> >> have a hard time, because they'd never imagine that
> >> someone who spoke fluent English would think that
> >> the two were different questions.
> >>
> > One is a stated question. The statement defines a coin flip sequence.
> > The other is a mathematical definition.
>
> > That's different.
>
> No. I'm sorry, but David is clearly right here and you are wrong.
> The presence or absence of the word "given" makes about as much
> difference as what colour of ink you use to print the question.
>
you define given. Take out the word given and the answer is 1/2. I'm
sorry, but that's true.
> > I say, and I'll say again. "Two coins were flipped and at least one is
> > a tail. What are the chances for two tails?" Has correct answer 1/2.
>
> On what planet? Flipping two coins can give the following results:
> HH, HT, TH, TT. We eliminate HH because it doesn't have at least one
> tail. The way we humans count, this leaves the probability of TT to
> be 1/3.
>
head is true." When there is TT, "at least one is a tail" is true.
With HT, or TH, either statement is true. The bettor betting for one
of each wins half the time. The bettor betting for two of a kind wins
half the time. That's fifty fifty.
Want to get a correct answer 1/3? You must alter the coin flip
sequence. You must reflip every fourth flip. You must decide this
prior to the flip. Your problem statement must, someway, somehow,
explain this reflip.
The word given does, or doesn't explain this reflip. It matters not to
me. Leave "given" out, and the correct answer is 1/2.
On this planet.
Eldon
Eldon Moritz
heads/tails decision must be made prior to the flip, you must reflip
every fourth flip. You must someway put this information into the
problem statement. On two of a kind, the losing bettor is asked to pay
two to one. The problem statement must explain to this bettor that the
same wouldn't have been true on the other two of a kind. Hmmmmmm......
Eldon
> - Randy
Eldon Moritz
> > I said something about what the asker of our question had to know. You
> > said "there was no asker." That's wrong. So far, you haven't
> > capitulated.We have a written question. It was written by someone, or
> > something.. Whatever asked the question, call whatever the asker.
>
> You have distorted, either deliberately or by misunderstanding,
> the discussion of the "asker".
>
thing is that I know nothing except what I learned from the statement.
The asker told me via the statement. That's elementary.
> Of course somebody wrote the problem. That means there is
> an author asking YOU, the reader something.
>
> The point of contention is whether a statement like
> "One of the coins is a heads" implies that the question
> author had to obtain this information by asking
> a question of SOMEBODY ELSE. Somebody who, in your
> interpretation, might have answered differently even
> if there was in fact a heads among the flips.
>
We know that our statement couldn't have been made without some kind
of investigation. Some way, some how, the writer of the question knew
that there were two coins. We also know that the writer found out the
outcome of one coin. In our question there is no evidence that the
writer knew the outcome of both coins.
That leaves us with correct answer 1/2.
> For some reason you accept that the question author
> can have test-givers omniscience when saying "given
> that one of the coins is a heads" but when the
> author leaves out the word "given" he loses his
> omniscience and must rely on polls and unreliable
> third parties.
>
I say that the mathematical statement the probability for two, given
at least one confuses folks. The math is worked out, and the answer is
1/3. It doesn't matter what the question was. There is a coin flip
sequence that would have this same hypothesis. It would have a reflip
every fourth flip. The question to correctly define this sequence
exists, or it doesn't, it doesn't matter. It isn't our statement, our
question.
The mathematics defines the hypothesis.
In our question, we have a question. It defines the mathematics. Go
directly to given at least one, every time you see "at least one is"
and you can never correctly answer our question.
I'm not arguing "given". I'm arguing our question. Don't let "given"
get in the way. I have asked what it means to say given. No one seems
to want to answer. Our question doesn't have given in it, so it's
moot. That's my argument.
Don't say our argument has the answer 1/3, because of what "given"
means. I didn't say the writer had to rely on third parties, when
given wasn't in the question. I said that the writer had to write a
true statement. To write a true statement the writer had to have had
some kind of inspection.
The writer can say "given" prior to the flip, I think. It depends upon
what it means.
On flips HH, the at least one is a head statement is true. With TT,
the "tails" statement is true. If we flipped with no prejudice, then
looked, we could make either statement, the bettors would bet fifty
fifty. We couldn't do this with "given" and get a 1/3. WE would have
to pick a side prior to the inspection.
Hope that clears it up.
Eldon
Joona I Palaste
First, you alter the premises, then you alter the question.
"Two coins are flipped and at least one is a tail. What is the
probability of two tails?"
That was the question.
Now you say it's equivalent to:
"Two coins are flipped. What is the probability of two of a kind?"
In case it might be too difficult for you, let me point out the
alterations.
You changed:
"Two coins are flipped and at least one is a tail" to
"Two coins are flipped"
and you changed:
"What is the probability of two tails?" to
"What is the probability of two of a kind?"
Both alterations change the answer to the question.
> Want to get a correct answer 1/3? You must alter the coin flip
> sequence. You must reflip every fourth flip. You must decide this
> prior to the flip. Your problem statement must, someway, somehow,
> explain this reflip.
No, it does not need to explain it. All it has to do is declare in
advance that two heads can never occur. For all the answerer knows,
there might be an omnipotent God preventing two heads from ever
occurring.
> The word given does, or doesn't explain this reflip. It matters not to
> me. Leave "given" out, and the correct answer is 1/2.
What the heck are you smoking? The answer to the question I wrote is
1/3 regardless of whether "given" appears or how you prevent two
heads from occurring.
> On this planet.
--
/-- Joona Palaste (pal...@cc.helsinki.fi) ---------------------------\
| Kingpriest of "The Flying Lemon Tree" G++ FR FW+ M- #108 D+ ADA N+++|
| http://www.helsinki.fi/~palaste W++ B OP+ |
\----------------------------------------- Finland rules! ------------/
"I will never display my bum in public again."
- Homer Simpson
David C. Ullrich
[...]
>>
>We know that our statement couldn't have been made without some kind
>of investigation.
No, we _don't_ know that. The question as stated is not about the
result of some investigation. It is a question about the probability
of something, _given_ certain information.
That's what the words _mean_. The fact that you disagree doesn't
matter, it's _still_ what the words mean.
[...]
>Hope that clears it up.
>
>Eldon
For a few days I've been wanting to comment that you've
never had any comment on certain remarks of Randy's.
Didn't comment because I didn't want to go to the trouble
of finding the post where they were made. Here it is.
You've never commented on the questions below.
Why not? (Because they indicate clearly that the
idea that we cannot be "given" something unless
the word "given" appears explicitly is ridiculous?)
>> Suppose I ask you another old chestnut: "On a mysterious
>> island you meet three natives. One always tells the
>> truth. One always lies. One alternately lies and
>> tells the truth...."
>>
>> The word "given" is absent. Do you then assume that
>> in fact my information about truth-telling and lie-telling
>> is unreliable?
>>
>> "A hen-and-a-half lays an egg-and-a-half in a day-and-a-half.
>> How many days does it take 10 hens to lay 10 eggs?"
>>
>> Again, the word "given" is absent. Thus, you feel you can't
>> take the first sentence as true for purposes of solving
>> the problem?
>>
>> Closer to home (I just made this up): There are 50 children
>> is Mrs. Smith's homeroom class. 15 have peanut-butter
>> sandwiches in their lunch today. 10 have apples...
>>
>> Do you assume that I can only have gleaned this information
>> by waiting for the children to volunteer the info? That
>> there might be more than 15 with peanut-butter sandwiches,
>> but only those 15 said something? That because I didn't
>> include the word "given", I didn't know it for sure when
>> I composed the problem about this fictitious class and
>> I am allowing this description to include classes where
>> the number is different from 15?
>>
>> - Randy
******************
David C. Ullrich
Eldon Moritz
know is what they hear in the statement. All they have to do to make
the proper bet is answer the question correctly. All we have to do to
figure their odds is answer the question correctly.
Bill always bets for two of a kind. Joe always bets for one of each.
When two coins were flipped they made one bet. You and I will agree
that's fifty fifty.
After they they hear the statement "at least one is a tail" they make
another bet. You and I would not agree here, I say they don't have
enough additional information to alter the second bet, that it's also
fifty fifty.
> In case it might be too difficult for you, let me point out the
> alterations.
> You changed:
> "Two coins are flipped and at least one is a tail" to
> "Two coins are flipped"
> and you changed:
> "What is the probability of two tails?" to
> "What is the probability of two of a kind?"
> Both alterations change the answer to the question.
>
I just added an additional step. The bettors made one bet when the
coins were flipped. Another after the statement.
They hear, "Two coins were flipped and at least on is a tail. What are
the chances for two tails?"
Bill bets for two of a kind. Joe bets for one of each. I didn't alter
the question, Bill and Joe just have a different way of expressing
their bet.
> > Want to get a correct answer 1/3? You must alter the coin flip
> > sequence. You must reflip every fourth flip. You must decide this
> > prior to the flip. Your problem statement must, someway, somehow,
> > explain this reflip.
>
> No, it does not need to explain it. All it has to do is declare in
> advance that two heads can never occur. For all the answerer knows,
> there might be an omnipotent God preventing two heads from ever
> occurring.
>
Declare it in advance. That's enough. I'll accept that. The bettors
have a right to know. It must be decided in advance, and the bettors
must be told. All they get is the statement. Put something in the
statement so the bettors will know. God can do it, just let God know
what you wish done. Also let the bettors know what God did.
> > The word given does, or doesn't explain this reflip. It matters not to
> > me. Leave "given" out, and the correct answer is 1/2.
>
> What the heck are you smoking? The answer to the question I wrote is
> 1/3 regardless of whether "given" appears or how you prevent two
> heads from occurring.
>
And on this, you and I disagree.
Eldon
> > On this planet.
Randy Poe
> rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
> > elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> > > I said something about what the asker of our question had to know. You
> > > said "there was no asker." That's wrong. So far, you haven't
> > > capitulated.We have a written question. It was written by someone, or
> > > something.. Whatever asked the question, call whatever the asker.
> >
> > You have distorted, either deliberately or by misunderstanding,
> > the discussion of the "asker".
> >
> No. We have a question. I'm the answerer. There was an asker. The main
> thing is that I know nothing except what I learned from the statement.
> The asker told me via the statement.
See? Now you're just lying, because in your very next paragraph
you introduce the third party again. The bone of contention
is the existence of this third party.
> > The point of contention is whether a statement like
> > "One of the coins is a heads" implies that the question
> > author had to obtain this information by asking
> > a question of SOMEBODY ELSE. Somebody who, in your
> > interpretation, might have answered differently even
> > if there was in fact a heads among the flips.
> >
> We know that our statement couldn't have been made without some kind
> of investigation.
Really? Then why do you accept "given that one of the coins
is a heads" without the investigation scenario?
If I'm writing a test, I'm creating the universe. I don't have
to investigate to know the facts in this universe. That's
elementary. If you are writing a test problem about
coin flips, you don't have to go flip some coins to write
it. You can make up hypothetical situations all you want in
your head, and you can establish the facts that you want
the test-taker to be aware of.
> Some way, some how, the writer of the question knew
> that there were two coins.
It's his story. He MADE IT UP. You deny an author's ability
to know the truth of HIS OWN STORY?
> We also know that the writer found out the
> outcome of one coin. In our question there is no evidence that the
> writer knew the outcome of both coins.
Except when the word "given" is present. Then he regains
his ability to know the facts of the story he is writing.
> > For some reason you accept that the question author
> > can have test-givers omniscience when saying "given
> > that one of the coins is a heads" but when the
> > author leaves out the word "given" he loses his
> > omniscience and must rely on polls and unreliable
> > third parties.
> >
> I say that the mathematical statement the probability for two, given
> at least one confuses folks.
It shouldn't. The part after the bar in a conditional probability
defines a subset of the universe. Define that subset correctly
and you can figure out which fraction has the event "two heads".
> I'm not arguing "given". I'm arguing our question. Don't let "given"
> get in the way. I have asked what it means to say given. No one seems
> to want to answer. Our question doesn't have given in it, so it's
> moot. That's my argument.
How can you think it's "moot" if it's absence changes the answer,
in your opinion? You're lying again. Obviously you think it
matters.
> Don't say our argument has the answer 1/3, because of what "given"
> means. I didn't say the writer had to rely on third parties, when
> given wasn't in the question.
You said the writer had to do an investigation, as if writers
are incapable of defining hypothetical situations.
> I said that the writer had to write a
> true statement. To write a true statement the writer had to have had
> some kind of inspection.
Or, as a writer, he could have MADE IT UP.
- Randy
Randy Poe
> heads/tails decision must be made prior to the flip,
Who said anything about "flipping two coins three ways"?
> you must reflip
> every fourth flip.
Another odd interpretation of conditional probability.
Conditional probability means sampling from a subset. In this
case, it means sampling from the subset of two-coin flips that
contain a heads. No pre-decision is needed. The decision is
made after the fact.
> You must someway put this information into the
> problem statement.
The information is already present in the problem statement
that we are interested in what proportion of flips that contain
one head, contains two heads.
Again you're making up extra scenarios that are not needed
or implied. Taking the problem-writer's scenario and altering it.
Here's another conditional probability scenario for you: I
do a random poll of people on the street about their shopping
habits. After I get home, my client wants to know what
proportion of people who shop at Target also shop at Walmart.
That is, "What is the probability X shops at Walmart given
X shops at Target?". In your view as stated above, I would
have had to pre-determined my sample and shot all the non-Target
people so they wouldn't contaminate the sample.
On our planet, since a conditional probability represents
a probability on a subset, all you have to do is select that subset
and you can do that after your experiment.
- Randy
Jason
this semantic nonsense.
Suppose two coins are flipped. They're behind a curtain so I can't see
them. Suppose first that I have a machine that inspects both coins for
heads or tails and reports this to my computer. The computer is
programmed to turn on a red light in case of both tails and a green
light otherwise. I turn on the machine and the green light comes on.
What do I know? I know at least one coin is heads. What, then, is the
probability that they're both heads? One third, of course.
Now suppose there's no machine. Rather, I reach back behind the
curtain and grab one of the coins. I look at it; it's heads. What do I
know? I know at least one coin is heads. Do I really know anything
more than this? I must, since now the probability that they're both
heads is one half. Conditional probabilities should depend only on the
given information, not on how that information was obtained. So what
else do I know besides, "at least one coin is heads"?
The "paradox" is resolved if the coins are distinguishable, say one
blue and one yellow. Then, in the second situation, either I know "the
blue coin is heads" or "the yellow coin is heads". Either piece of
information is more than just knowing that "at least one coin is
heads". So to resolve the paradox in general, should we suppose the
coins are distinguishable once I've picked one? I.e., instead of "blue
coin" and "yellow coin", there's "coin I picked" and "coin I didn't
pick"?
Or, prehaps it's more prudent to consider all distinct objects to be
distinguishable (else they'd be the same object) and that the concept
of "indistinguishable" is simply a fictitious mathematical
simplification, which in this case simplifies too much and throws away
some information.
David C. Ullrich
Yes.
>Or, prehaps it's more prudent to consider all distinct objects to be
>distinguishable (else they'd be the same object) and that the concept
>of "indistinguishable" is simply a fictitious mathematical
>simplification, which in this case simplifies too much and throws away
>some information.
There's no paradox - you're making an error. Well, actually you
point out the resolution of what seems to you to be a paradox,
so I don't see why you think it's paradoxical...
If you know exactly that at leasy one coin is a head then
the probability that both are heads is 1/3. If you reach behind
the curtain and pick one, and it turns out to be a head, then
the probability that both are heads is 1/2. The reason there's
no paradox to be resolved is that reaching behind the curtain,
picking one, and seeing that it's a head is simply not the
same thing as knowing somehow that precisely one is
a head - it tells you that _the one you picked_ is a head.
In the first case we know somehow _precisely_ that
one of the two coins is a head. For example we could
be, ahem, _given_ this information in the statement
of the problem, or we could assume there's a man
behind the curtain who always tells the truth, and
we asked him "is at least one a head?". Asking that
man that question is a way to determine precisely
whether or not one is a head, no more and no
less. On the other hand reaching behind the
curtain and looking at one of the coins is
_not_ a valid way to determine precisely whether
or not one is a head - if you do that and you see
a head you get more information than the answer
to the question "is at least one a head", while if you
do that and you see a tail you get less information
than that.
******************
David C. Ullrich
Eldon Moritz
> > Two coins flip four ways. To flip two coins three ways, the
> > heads/tails decision must be made prior to the flip,
>
> Who said anything about "flipping two coins three ways"?
>
flipped two coins, three ways. Flip two coins, you get fourths. To get
a correct answer of 1/3, you must change this to thirds.
> > you must reflip
> > every fourth flip.
>
> Another odd interpretation of conditional probability.
>
TH, TT. In other words, flip two coins and you get fourths. To get
thirds, you must eliminate one, prior to the flip.
> Conditional probability means sampling from a subset. In this
> case, it means sampling from the subset of two-coin flips that
> contain a heads. No pre-decision is needed. The decision is
> made after the fact.
>
coin flips about which the "at least one is a heads" statement was
made.
You are sampling from the subset of which our statement would have
been true. This changes the question. Our statement WAS made. It's a
historical event. It happened.
> > You must someway put this information into the
> > problem statement.
>
> The information is already present in the problem statement
> that we are interested in what proportion of flips that contain
> one head, contains two heads.
>
> Again you're making up extra scenarios that are not needed
> or implied. Taking the problem-writer's scenario and altering it.
>
> Here's another conditional probability scenario for you: I
> do a random poll of people on the street about their shopping
> habits. After I get home, my client wants to know what
> proportion of people who shop at Target also shop at Walmart.
> That is, "What is the probability X shops at Walmart given
> X shops at Target?". In your view as stated above, I would
> have had to pre-determined my sample and shot all the non-Target
> people so they wouldn't contaminate the sample.
>
Different analogy entirely.
> On our planet, since a conditional probability represents
> a probability on a subset, all you have to do is select that subset
> and you can do that after your experiment.
>
Yes, you can, but you have to determine the proper subset.
***You cannot, on this planet, change "fourths" to "thirds", after you
look at the toss.*** That's a challenge. If you can do it, I'll
capitulate.
Eldon
> - Randy
Eldon Moritz
> There seems to be a genuinely interesting "paradox" lurking behind all
> this semantic nonsense.
>
> Suppose two coins are flipped. They're behind a curtain so I can't see
> them. Suppose first that I have a machine that inspects both coins for
> heads or tails and reports this to my computer. The computer is
> programmed to turn on a red light in case of both tails and a green
> light otherwise. I turn on the machine and the green light comes on.
> What do I know? I know at least one coin is heads. What, then, is the
> probability that they're both heads? One third, of course.
>
turn on two different colored lights.
Suppose that we see a red and a blue, then we say "at least one is a
blue?"
Then the computer said, "Two coins were flipped and at least one is a
tail. What are the chances for two tails?"
Would we both answer that 1/2?
Eldon
Kevin Buhr
>
> Honest. I point this out expliictly for the benefit of readers
> who might try to figure out what his point is, and might
> have a hard time, because they'd never imagine that
> someone who spoke fluent English would think that
> the two were different questions.
I'm chiming in a little late, but I think that this misrepresents the
issue.
A nonmathematician who speaks English fluently might never imagine
they were two different questions but might not get the "right"
answer or even understand the point, so who cares?
A mathematician knows that "given X what is the chance of Y?" is
asking for a conditional probability, something mathematically well
defined. That mathematician *might* conclude that, since the two
English questions are so similar, the first must have the same meaning
as the second and thus must *also* be asking for the same conditional
probability. It's certainly reasonable, but I believe it requires
making allowances for the sloppy way in which the first question is
posed.
Look, when we say "two coins were flipped", by convention every
mathematician knows that we're talking about the sample space
Omega={HH,HT,TH,TT} with probability P_0 putting mass 1/4 on each
point. Maybe this is more than a convention, and maybe it isn't.
(For example, when we say "a countably infinite number of coins were
flipped", it becomes less clear---at least to me---that the understood
probability space is any more than just a shared convention.)
But what does it mean when we say "two coins were flipped and at least
one is a head"? You, apparently, think the appropriate convention is
to take it to mean "the probability space is (Omega,P_0) the same as
before, but from now on whenever I ask for an apparently unconditional
probability, I'm really asking for a conditional probability given
this event I just described."
I believe Eldon thinks it means "the probability space (Omega,P_0) is
the same as before, and some random observational process, the details
of which have been omitted from the statement of the question, occurs
that allows us to conclude that omega is in {HT,TH,TT}; also, from now
on whenever I ask for an apparently unconditional probability, I'm
really asking for a conditional probability given the result of this
random observational process."
[[For the record, I think both interpretations are incorrect. I think
the correct mathematical interpretation of this poorly posed question
is that the sample space is Omega'={HT,TH,TT}, the appropriate
probability on it has to be assumed but a good convention *might be*
to take it to be P_1(omega) = P_0(omega|Omega'), and I get the same
answers as you do, but I happen to belong to the minority that
believes the two questions are actually asking different things and
that the first, it could be argued, is not well posed.]]
I believe Eldon further reasons that the structure of the question
implies that the random observational process always occurs---it would
have occured even in the case where omega was HH. Obviously, in this
case, it couldn't have given us the information that omega was in
{HT,TH,TT}, so the observational process must be a little more
complicated than "always say omega is in {HT,TH,TT}". He concludes
that, in the absence of other information, it's reasonable to assume
that the observational process is something like: examine the result
of both flips, and if you don't have a choice, say there's at least
one head or tail as appropriate; if you do have a choice, pick a coin
at random and say there's at least one of whatever that one is.
He would say that the alternative would be to assume that the random
observational process says "there is at least one tail" whenever there
is at least one tail, and says "there is at least one head" or "both
coins are heads" or "ERROR ERROR does not compute SHUTTING DOWN
UNIVERSE IN 3...2...1..." whenever there are two heads. And it's
silly to, a priori, assume the observational process has a bias
towards observing tails.
> It's very curious. He keeps asking me to explain what's
> wrong with what he says, but when I do he never seems
> to realize that I'm simply doing so, instead we get stuff
> like "Methinks the perfesser thought that shit up to protect
> his ignorant position." Hence the phrase "dumb fucking
> asshole".
It's possible he's just a jerk, but it seems more likely that---like
you---he's totally and completely convinced he's right but, unlike
you, doesn't have the mathematical background to articulate his
argument in the mathematical terms that you, and others here, will
find convincing.
--
Kevin <bu...@telus.net>
Jason
I'm not so sure the "yes" goes here. But more on that below.
>
> >Or, prehaps it's more prudent to consider all distinct objects to be
> >distinguishable (else they'd be the same object) and that the concept
> >of "indistinguishable" is simply a fictitious mathematical
> >simplification, which in this case simplifies too much and throws away
> >some information.
>
> There's no paradox - you're making an error. Well, actually you
> point out the resolution of what seems to you to be a paradox,
> so I don't see why you think it's paradoxical...
Well, of course, I don't believe there's any mathematical paradox.
It's more a paradox of language if anything. At any rate, that's why
paradox was originally in quotes.
>
> If you know exactly that at leasy one coin is a head then
> the probability that both are heads is 1/3. If you reach behind
> the curtain and pick one, and it turns out to be a head, then
> the probability that both are heads is 1/2. The reason there's
> no paradox to be resolved is that reaching behind the curtain,
> picking one, and seeing that it's a head is simply not the
> same thing as knowing somehow that precisely one is
> a head - it tells you that _the one you picked_ is a head.
>
> In the first case we know somehow _precisely_ that
> one of the two coins is a head. For example we could
> be, ahem, _given_ this information in the statement
> of the problem, or we could assume there's a man
> behind the curtain who always tells the truth, and
> we asked him "is at least one a head?". Asking that
> man that question is a way to determine precisely
> whether or not one is a head, no more and no
> less. On the other hand reaching behind the
> curtain and looking at one of the coins is
> _not_ a valid way to determine precisely whether
> or not one is a head - if you do that and you see
> a head you get more information than the answer
> to the question "is at least one a head",
This is my point. What, precisely, is the additional information?
We've both suggested that the additional information is that "_the one
you picked_ is a head". Well, if the coins are genuinely
indistinguishable, it can be legitimately argued that this isn't
additional information at all.
That's why I say it's a language-related paradox. Try to explain
without mathematics how "_the one you picked_ is a head" says anything
more than "at least one is a head", under the (non-mathematical)
assumption that the coins are "indistinguishable".
This leads me back to why I don't think the "yes" goes where you put
it. I think it goes one paragraph down. I can't conceive of any way to
set up the probability space in a mathematical model of this situation
without implicitly assuming some distinguishability. The probability
space has to be rich enough to model both experiments: the flipping of
the two coins and the picking one from behind the curtain, so
\Omega={HH,TH,TT} with P({HH})=P({TT})=1/4 and P({TH})=1/2 won't work.
If we use the simple \Omega={HH,HT,TH,TT}, we're implying there's a
first coin and a second coin. In any valid setup, the coins are going
to be distinguishable, from the beginning, even before I pick one.
And, after all, even in plain language, why shouldn't they be? They're
not the same coin. If I hold them in my hand, I'm not fooled into
thinking I'm holding only one coin! :)
Virgil
<349f5619.03040...@posting.google.com>,
elmo...@yahoo.com (Eldon Moritz) wrote:
> When two coins are flipped, there are two alike on each end. HH, HT,
> TH, TT. In other words, flip two coins and you get fourths. To get
> thirds, you must eliminate one, prior to the flip.
Not necessarily. You can do it during the flipping by
declaring that one of the 4 results result is not an
acceptable. In effect, you simply repeat the experiment
until you get an acceptable result.
Thus the experiment might be described as flipping two
coins until at least one is heads.
The probability of nothing but a sequence of double tails
decreases rapidly as the number of repetitions increases,
less than 1/1,000 for 5 flips or 1/1,000,000 for 10.
Kevin Buhr
>
> It's possible he's just a jerk, but it seems more likely that---like
> you---he's totally and completely convinced he's right but, unlike
> you, doesn't have the mathematical background to articulate his
> argument in the mathematical terms that you, and others here, will
> find convincing.
Actually, I should add that it's by no means clear that there *is* a
way of articulating his argument in such terms. It seems likely that
his interpretation can't be made consistent over all similar
questions.
For example, I'd be interested to know how he'd interpret this
question:
Five coins are flipped, at least one is a tail, and at least
two are heads. What is the probability that there are three
tails?
If I understand correctly, Eldon would want to assume some sort of
unbiased, omniscient asker/tester gave us this information, but it's
not clear what "rule" this tester is following. And if it's not clear
here, why is it clear in the two-coin question?
--
Kevin <bu...@telus.net>
Michael Hochster
: The "paradox" is resolved if the coins are distinguishable, say one
: blue and one yellow. Then, in the second situation, either I know "the
: blue coin is heads" or "the yellow coin is heads". Either piece of
: information is more than just knowing that "at least one coin is
: heads". So to resolve the paradox in general, should we suppose the
: coins are distinguishable once I've picked one? I.e., instead of "blue
: coin" and "yellow coin", there's "coin I picked" and "coin I didn't
: pick"?
Yes. Imagine that both coins start out yellow and that,
after picking one, you paint it blue.
Mike
David C. Ullrich
>rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
>> elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
>> > Two coins flip four ways. To flip two coins three ways, the
>> > heads/tails decision must be made prior to the flip,
>>
>> Who said anything about "flipping two coins three ways"?
>>
>When you flip two coins with three equally likely outcomes, you have
>flipped two coins, three ways. Flip two coins, you get fourths. To get
>a correct answer of 1/3, you must change this to thirds.
>
>> > you must reflip
>> > every fourth flip.
>>
>> Another odd interpretation of conditional probability.
>>
>When two coins are flipped, there are two alike on each end. HH, HT,
>TH, TT. In other words, flip two coins and you get fourths. To get
>thirds, you must eliminate one, prior to the flip.
>
>> Conditional probability means sampling from a subset. In this
>> case, it means sampling from the subset of two-coin flips that
>> contain a heads. No pre-decision is needed. The decision is
>> made after the fact.
>>
>Here we differ. In this case it means sampling from the subset of two
>coin flips about which the "at least one is a heads" statement was
>made.
_If_ there was some mention of a statement being made in
"A coin is tossed twice. At least one is a head. What is the
probability that there are two heads?" _then_ that is what
it would mean. But there is _no_ mention of any statement
being made in that statement. It refers to the set of pairs
of flips about which the statement "at least one is a head"
_is_ _true_.
******************
David C. Ullrich
David C. Ullrich
>David C. Ullrich <ull...@math.okstate.edu> writes:
>>
>> Honest. I point this out expliictly for the benefit of readers
>> who might try to figure out what his point is, and might
>> have a hard time, because they'd never imagine that
>> someone who spoke fluent English would think that
>> the two were different questions.
>
>I'm chiming in a little late, but I think that this misrepresents the
>issue.
>
>A nonmathematician who speaks English fluently might never imagine
>they were two different questions but might not get the "right"
>answer or even understand the point, so who cares?
>
>A mathematician knows that "given X what is the chance of Y?" is
>asking for a conditional probability, something mathematically well
>defined. That mathematician *might* conclude that, since the two
>English questions are so similar, the first must have the same meaning
>as the second and thus must *also* be asking for the same conditional
>probability. It's certainly reasonable, but I believe it requires
>making allowances for the sloppy way in which the first question is
>posed.
>
>
>
>Look, when we say "two coins were flipped", by convention every
>mathematician knows that we're talking about the sample space
>Omega={HH,HT,TH,TT} with probability P_0 putting mass 1/4 on each
>point. Maybe this is more than a convention, and maybe it isn't.
The interpretation of any sequence of English words is ultimately
based on nothing but convention - the fact that "dog" means dog
and not cat is just a convention. The question is not whether the
thing is a matter of convention, the question is what the convention
_is_.
>(For example, when we say "a countably infinite number of coins were
>flipped", it becomes less clear---at least to me---that the understood
>probability space is any more than just a shared convention.)
>
>But what does it mean when we say "two coins were flipped and at least
>one is a head"? You, apparently, think the appropriate convention is
>to take it to mean "the probability space is (Omega,P_0) the same as
>before, but from now on whenever I ask for an apparently unconditional
>probability, I'm really asking for a conditional probability given
>this event I just described."
>
>I believe Eldon thinks it means "the probability space (Omega,P_0) is
>the same as before, and some random observational process, the details
>of which have been omitted from the statement of the question, occurs
>that allows us to conclude that omega is in {HT,TH,TT}; also, from now
>on whenever I ask for an apparently unconditional probability, I'm
>really asking for a conditional probability given the result of this
>random observational process."
>
>[[For the record, I think both interpretations are incorrect. I think
>the correct mathematical interpretation of this poorly posed question
>is that the sample space is Omega'={HT,TH,TT}, the appropriate
>probability on it has to be assumed but a good convention *might be*
>to take it to be P_1(omega) = P_0(omega|Omega'), and I get the same
>answers as you do,
Hence I don't see the point to the distinction.
>but I happen to belong to the minority that
>believes the two questions are actually asking different things and
>that the first, it could be argued, is not well posed.]]
Yes, it's certainly true that the "at least one is a head" is not
worded very well - in a formal mathematical context I'd say it
needed to be restated. But this sort of wording is very common
in informal descriptions of puzzles - when we read "One and a
half chickens can lay one and a half eggs in one and a half
days..." we don't assume that the writer is actually asserting
something that's supposed to be _true_, we understand that
he's giving us hypotheses, even though the gramattical
markers indicating that they are hypotheses have been
omitted.
In particular we _don't_ start analyzing the problem on
the basis of how we obtained the information; on the
basis of who said this and why...
>I believe Eldon further reasons that the structure of the question
>implies that the random observational process always occurs
Well duh, yes it's clear that this is what he thinks. But:
>---it would
>have occured even in the case where omega was HH. Obviously, in this
>case, it couldn't have given us the information that omega was in
>{HT,TH,TT}, so the observational process must be a little more
>complicated than "always say omega is in {HT,TH,TT}". He concludes
>that, in the absence of other information, it's reasonable to assume
>that the observational process is something like: examine the result
>of both flips, and if you don't have a choice, say there's at least
>one head or tail as appropriate; if you do have a choice, pick a coin
>at random and say there's at least one of whatever that one is.
As I've pointed out many times, _if_ we take the question that
way then there's no way to give any sort of answer - how did we
determine that the above is the "reasonable" assumption?
>He would say that the alternative would be to assume that the random
>observational process says "there is at least one tail" whenever there
>is at least one tail, and says "there is at least one head" or "both
>coins are heads" or "ERROR ERROR does not compute SHUTTING DOWN
>UNIVERSE IN 3...2...1..." whenever there are two heads. And it's
>silly to, a priori, assume the observational process has a bias
>towards observing tails.
>
>> It's very curious. He keeps asking me to explain what's
>> wrong with what he says, but when I do he never seems
>> to realize that I'm simply doing so, instead we get stuff
>> like "Methinks the perfesser thought that shit up to protect
>> his ignorant position." Hence the phrase "dumb fucking
>> asshole".
>
>It's possible he's just a jerk, but it seems more likely that---like
>you---he's totally and completely convinced he's right but, unlike
>you, doesn't have the mathematical background to articulate his
>argument in the mathematical terms that you, and others here, will
>find convincing.
That's not how I see it. Seems to me that he's simply interpreting
the question incorrectly. Now, I agree that the fact that he's
interpreting the question incorrectly has something to do with
his background, but I don't think the problem is that he doesn't
have the ability to articulate his arguments coherently. It seems
to me that he's simply misunderstanding the meaning of a
certain informal English construction - I haven't given any
"mathematical" arguments that it is what I say it is, because
that's not a mathematical question.
It's a question that has a lot to do with mathematics, but
it's not a mathematical question in the sense that it's
susceptible to mathematical proof or disproof.
******************
David C. Ullrich
David C. Ullrich
Precisely. I've said many times that _if_ we interpret things in
the sort of way that he's interpreting them then there's no way
to say what the answer is, because we simply have no information
about relevant details - those details being what you call the "rule"
here.
On the other hand, interpreting the question as meaning
"given that at least one is a head" (or if you prefer, "restricting
our sample space to HH, TH, HT", not that I see why the
difference matters) _is_ a reasonable interpretation. Why
is that a reasonable interpretation? Because that is the
way people _do_ interpret the question! And whoever
wrote the question presumably knows that people
will interpret it that way - hence he would have written
something else if he meant something else.
******************
David C. Ullrich
David C. Ullrich
Perhaps that could be "legitimately" argued, but I don't
think it could be correctly argued.
It's clear that painting one coin red and one coin blue
has no effect on the probabilities. But if we paint the
coins then it's clear that we do obtain additional
information.
>That's why I say it's a language-related paradox. Try to explain
>without mathematics how "_the one you picked_ is a head" says anything
>more than "at least one is a head", under the (non-mathematical)
>assumption that the coins are "indistinguishable".
>
>This leads me back to why I don't think the "yes" goes where you put
>it. I think it goes one paragraph down. I can't conceive of any way to
>set up the probability space in a mathematical model of this situation
>without implicitly assuming some distinguishability. The probability
>space has to be rich enough to model both experiments: the flipping of
>the two coins and the picking one from behind the curtain, so
>\Omega={HH,TH,TT} with P({HH})=P({TT})=1/4 and P({TH})=1/2 won't work.
>If we use the simple \Omega={HH,HT,TH,TT}, we're implying there's a
>first coin and a second coin. In any valid setup, the coins are going
>to be distinguishable, from the beginning, even before I pick one.
>And, after all, even in plain language, why shouldn't they be? They're
>not the same coin. If I hold them in my hand, I'm not fooled into
>thinking I'm holding only one coin! :)
>
>> while if you
>> do that and you see a tail you get less information
>> than that.
>>
>> ******************
>>
>> David C. Ullrich
******************
David C. Ullrich
Randy Poe
> this semantic nonsense.
>
> Suppose two coins are flipped. They're behind a curtain so I can't see
> them. Suppose first that I have a machine that inspects both coins for
> heads or tails and reports this to my computer. The computer is
> programmed to turn on a red light in case of both tails and a green
> light otherwise. I turn on the machine and the green light comes on.
> What do I know? I know at least one coin is heads. What, then, is the
> probability that they're both heads? One third, of course.
>
> Now suppose there's no machine. Rather, I reach back behind the
> curtain and grab one of the coins. I look at it; it's heads. What do I
> know?
This is the "one child is a boy" experiment (but not Eldon's
version of it).
> I know at least one coin is heads. Do I really know anything
> more than this? I must, since now the probability that they're both
> heads is one half. Conditional probabilities should depend only on the
> given information, not on how that information was obtained.
But "how that information was obtained" IS part of the given
experiment. In the language of probability, the event
described by "one coin is a head" is not the same event in
both cases. It is not the same subset of outcome space.
The reason is that the whole description of the event includes
the information on how the information was obtained.
> So what
> else do I know besides, "at least one coin is heads"?
>
> The "paradox" is resolved if the coins are distinguishable, say one
> blue and one yellow. Then, in the second situation, either I know "the
> blue coin is heads" or "the yellow coin is heads". Either piece of
> information is more than just knowing that "at least one coin is
> heads". So to resolve the paradox in general, should we suppose the
> coins are distinguishable once I've picked one? I.e., instead of "blue
> coin" and "yellow coin", there's "coin I picked" and "coin I didn't
> pick"?
Yes. That's good language to distinguish the two events.
Experiment 1: One coin is a heads or (the computer sees that
one coin is a heads).
Experiment 2: The coin you picked randomly is a heads.
That makes it clearer these are not the same outcomes.
- Randy
Eldon Moritz
> Kevin Buhr <bu...@telus.net> writes:
Kevin, I'm glad you appeared. You're the first who has shown lately
with mathematical expertise who is willing to analyze this question.
The rest just make a couple of assumptions, then bypass my argument
and go on.
> >
> > It's possible he's just a jerk, but it seems more likely that---like
> > you---he's totally and completely convinced he's right but, unlike
> > you, doesn't have the mathematical background to articulate his
> > argument in the mathematical terms that you, and others here, will
> > find convincing.
>
This is a given. Eldon is a jerk; he admits it. This doesn't alter the
validity of his argument. Eldon's mathematical background is limited.
His logic is superb. He had some college math and understands the
Either, Or's.
I'm studying your previous post, and will submit a post, answering, or
articulating on it. I think I agree with most of it.
> Actually, I should add that it's by no means clear that there *is* a
> way of articulating his argument in such terms. It seems likely that
> his interpretation can't be made consistent over all similar
> questions.
>
My basic argument is that I don't disagree that P ( A | B ) means that
P is the probability for A happening, given that B has actually
happened. I agree with the mathematical derivations of B.
My difference with the dissenters in this post has been what actually
is B? What actually happened.
In the question "Two coins were flipped and at least one is a head.
What are the chances for two heads?"
B is, what actually happened is, the above statement was made. All we
know about the flip, we learned from the above statement. Had there
not been a statement, we would not even have known there was a flip.
> For example, I'd be interested to know how he'd interpret this
> question:
>
> Five coins are flipped, at least one is a tail, and at least
> two are heads. What is the probability that there are three
> tails?
>
Same way. What is the probability for three tails, given that we were
told "Five coins are flipped, at least one is a tail, and at least two
are heads." That's all we know.
> If I understand correctly, Eldon would want to assume some sort of
> unbiased, omniscient asker/tester gave us this information, but it's
> not clear what "rule" this tester is following. And if it's not clear
> here, why is it clear in the two-coin question?
In the two-coin question, my working model works. In the above
question, can I make a working model work? Can you? Does it matter?
When two coins are flipped one of four things happens (HH, or HT, or
TH, or TT). Notice the or's. They can happen in any order. It is a
fact that our statement could not have been made, prior to the flip.
Suppose that HH happened and our statement was made. Bill bets for two
H's, Joe bets for one of each. Bill will win. What odds should Joe
pay?
Suppose that TT happened and the "at least one is a tails" statement
was made. Would that question have a different answer? Bill would bet
for two tails, Bill would win. Should Joe pay two to one?
At TH, and HT, either statement would be true. Bill lose, Joe would
win. Would it matter which statement was made?
About the five card flip. If we can't make the model for the latter,
does that negate the former?
In the two-coin flip, someone well versed in conditional probability
should be able to answer, given that the statement was made, what are
the chances that it was made from HT, versus what are the chances that
it was made from HH.
When you get to a five coin flip, the question gets a little tougher,
but given that the above statement was made, someone deep into
conditional probability should be able to crack it. (I'm probably not
in that deep)
A mathematician would, or should, first identify the givens. First,
clearly identify what the problem is.
I'm a jerk, but I'm a lovable little jerk.
Eldon
Randy Poe
> rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
> > elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> > > Two coins flip four ways. To flip two coins three ways, the
> > > heads/tails decision must be made prior to the flip,
> >
> > Who said anything about "flipping two coins three ways"?
> >
> When you flip two coins with three equally likely outcomes, you have
> flipped two coins, three ways. Flip two coins, you get fourths. To get
> a correct answer of 1/3, you must change this to thirds.
By selecting a subset of flips after the fact.
> When two coins are flipped, there are two alike on each end. HH, HT,
> TH, TT. In other words, flip two coins and you get fourths. To get
> thirds, you must eliminate one, prior to the flip.
Or after.
> > Conditional probability means sampling from a subset. In this
> > case, it means sampling from the subset of two-coin flips that
> > contain a heads. No pre-decision is needed. The decision is
> > made after the fact.
> >
> Here we differ. In this case it means sampling from the subset of two
> coin flips about which the "at least one is a heads" statement was
> made.
Which can be made after, by looking at both flips. Your
error is in assuming that I wouldn't tell you one is a heads
after I look at both. That I might always choose to answer
something differently, even if I tell you EXPLICITLY I
am never going to say "one is a heads". I tell outright
that I am going to look at the flips and make either the
statement "one is a heads" or "there are no heads", and
no other statement.
> You are sampling from the subset of which our statement would have
> been true.
Right. Now, of which subset of flips is the statement "one
is a heads" true. Not "did a hypothetical and mischievous
observer happen to say one is a heads" but "one is a heads".
Look at this flip: HT. Is the statement "one is a heads" true?
If so, I include it in my subset. Always.
Look at this flip: TH. Is the statement "one is a heads" true?
If so, I include it in my subset. Always.
Look at this flip: HH. Is the statement "one is a heads" true?
If so, I include it in my subset. Always.
Look at this flip: TT. Is the statement "one is a heads" true?
If not, I do not include it in my subset. Ever.
As you say, the subset is that subset of flips, after the
fact, for which the STATEMENT is true.
> > Here's another conditional probability scenario for you: I
> > do a random poll of people on the street about their shopping
> > habits. After I get home, my client wants to know what
> > proportion of people who shop at Target also shop at Walmart.
> > That is, "What is the probability X shops at Walmart given
> > X shops at Target?". In your view as stated above, I would
> > have had to pre-determined my sample and shot all the non-Target
> > people so they wouldn't contaminate the sample.
> >
> Different analogy entirely.
Nope. Same concept: Proportion of a subset, which can be
chosen after the fact.
> > On our planet, since a conditional probability represents
> > a probability on a subset, all you have to do is select that subset
> > and you can do that after your experiment.
> >
> Yes, you can, but you have to determine the proper subset.
>
> ***You cannot, on this planet, change "fourths" to "thirds", after you
> look at the toss.*** That's a challenge. If you can do it, I'll
> capitulate.
Subset. Subset. Subset. What is so hard about taking a subset?
Think about my polling example. I can indeed change from
"proportion of all shoppers who shop at Walmart" to "proportion
of Target shoppers who shop at Walmart" and I can do it
after the fact. It's EASY!!!
I do it this way: I'm keeping a tally sheet with two columns.
One is total number of two-flip tosses that include a heads,
the other is flips with two heads. Every time heads comes
up at least once, I make a mark in column A. When it comes
up twice, I make a mark in column B. When two tails comes
up, I make no mark.
At the end of the experiment, I divide the number of marks
in column B by the number of marks in column A.
I can even write a simple computer program designed to do
this, and it doesn't have to prejudice the flips beforehand.
It just has to do this (pseudo-code):
flip = random_coin_toss();
if (flip == 'HT') {colA++;}
else if (flip == 'HH') {colA++; colB++}
else {} /* Two tails: do nothing */
Does that bother you? How about this?
flip = random_coin_toss();
if (flip == 'HT') {colA++; total++}
else if (flip == 'HH') {colA++; colB++; total++}
else {total++;}
There. Now I'm counting every toss that occurs (in
"total"), throwing out nothing. But colA is only
about 3/4 of total, and colB is about 1/3 of colA,
or 1/4 of total.
I chose my subset after the flip, just as I said. Why is
that such a hard concept?
- Randy
Randy Poe
> >
> > It's possible he's just a jerk, but it seems more likely that---like
> > you---he's totally and completely convinced he's right but, unlike
> > you, doesn't have the mathematical background to articulate his
> > argument in the mathematical terms that you, and others here, will
> > find convincing.
>
> Actually, I should add that it's by no means clear that there *is* a
> way of articulating his argument in such terms. It seems likely that
> his interpretation can't be made consistent over all similar
> questions.
>
> For example, I'd be interested to know how he'd interpret this
> question:
>
> Five coins are flipped, at least one is a tail, and at least
> two are heads. What is the probability that there are three
> tails?
>
> If I understand correctly, Eldon would want to assume some sort of
> unbiased, omniscient asker/tester gave us this information,
No, he wants us to assume we have a very limited asker/tester
here who has to rely on someone else. That someone else
might not say "at least one is a tail" for all cases in
which at least one is a tail.
- Randy
David C. Ullrich
wrote:
>Kevin Buhr <bu...@telus.net> wrote in message news:<87u1d7q...@saurus.asaurus.invalid>...
>> Kevin Buhr <bu...@telus.net> writes:
>
>Kevin, I'm glad you appeared. You're the first who has shown lately
>with mathematical expertise who is willing to analyze this question.
>The rest just make a couple of assumptions, then bypass my argument
>and go on.
>
>> >
>> > It's possible he's just a jerk, but it seems more likely that---like
>> > you---he's totally and completely convinced he's right but, unlike
>> > you, doesn't have the mathematical background to articulate his
>> > argument in the mathematical terms that you, and others here, will
>> > find convincing.
>>
>This is a given. Eldon is a jerk; he admits it. This doesn't alter the
>validity of his argument. Eldon's mathematical background is limited.
>His logic is superb. He had some college math and understands the
>Either, Or's.
He also referrs to himself in the third person. Strange.
>I'm studying your previous post, and will submit a post, answering, or
>articulating on it. I think I agree with most of it.
Including the assertion that the answer to A' and B' is the same?
>> Actually, I should add that it's by no means clear that there *is* a
>> way of articulating his argument in such terms. It seems likely that
>> his interpretation can't be made consistent over all similar
>> questions.
>>
>My basic argument is that I don't disagree that P ( A | B ) means that
>P is the probability for A happening, given that B has actually
>happened. I agree with the mathematical derivations of B.
>
>My difference with the dissenters in this post has been what actually
>is B? What actually happened.
>
>In the question "Two coins were flipped and at least one is a head.
>What are the chances for two heads?"
>
>B is, what actually happened is, the above statement was made.
No. What we're _given_, what actually _happened_, is that
two coins were flipped and at least one was a head. All this
stuff about who said what, when and why is just your invention.
Really? That's very curious. You've stated in public that you're
one of the foremost experts, perhaps _the_ expert, on the
original question. But you're probably not able to figure out
the answer to a question that's just a teensy bit more complicated?
That's very curious.
>A mathematician would, or should, first identify the givens. First,
>clearly identify what the problem is.
Uh, yes. The fact that the mathematicians do not agree with
you on what the givens are, perversely insisting that they
are exactly what's given, doesn't show that they don't
agree that one should first identify what the givens are.
>I'm a jerk, but I'm a lovable little jerk.
Possibly - you've given no evidence of it here.
>Eldon
******************
David C. Ullrich
Michael J. Doré
Maybe a semi-reasonable interpretation is this "provider of
information" picks a subset of the five coins (selects each coin with
probability 1/2, independently of each other and independently of
whether they are heads or tails) and then tells us the number of heads
and tails on his/her selection. This is consistent with Eldon's answer
of 1/2 in the simpler example.
However I agree with those who say this is reading far more into the
question than what's there - by far the most natural interpretation is
surely to condition on exactly what we're given, and not worry about
how we got the information (since this is not referred to in the
question).
Of course it would be different if the question had been: two coins
are tossed, someone sees the coins, we ask that person "tell us the
the result of one of the coin tosses" and (s)he answers "heads". Now
the probability of two heads is 1/2 under the most reasonable
interpretations (which involve assuming the person we're asking has no
heads/tails bias).
Michael
Eldon Moritz
> On Thu, 10 Apr 2003 02:09:21 GMT, Kevin Buhr <bu...@telus.net> wrote:
>
> >Kevin Buhr <bu...@telus.net> writes:
> >>
> >> It's possible he's just a jerk, but it seems more likely that---like
> >> you---he's totally and completely convinced he's right but, unlike
> >> you, doesn't have the mathematical background to articulate his
> >> argument in the mathematical terms that you, and others here, will
> >> find convincing.
> >
> >Actually, I should add that it's by no means clear that there *is* a
> >way of articulating his argument in such terms. It seems likely that
> >his interpretation can't be made consistent over all similar
> >questions.
> >
> >For example, I'd be interested to know how he'd interpret this
> >question:
> >
> > Five coins are flipped, at least one is a tail, and at least
> > two are heads. What is the probability that there are three
> > tails?
> >
> >If I understand correctly, Eldon would want to assume some sort of
> >unbiased, omniscient asker/tester gave us this information, but it's
> >not clear what "rule" this tester is following. And if it's not clear
> >here, why is it clear in the two-coin question?
>
which was "told" to us by the statement.
> Precisely. I've said many times that _if_ we interpret things in
> the sort of way that he's interpreting them then there's no way
> to say what the answer is, because we simply have no information
> about relevant details - those details being what you call the "rule"
> here.
>
show you how to get the correct answer. The details? All the details
we have is what we got from the statement.
What the statement told us. That's all we know!!!!! All the relevant
details ARE in our statement. Our statement is complete.
> On the other hand, interpreting the question as meaning
> "given that at least one is a head" (or if you prefer, "restricting
> our sample space to HH, TH, HT", not that I see why the
> difference matters) _is_ a reasonable interpretation.
It may be reasonable, but it gets the wrong answer. It adds
information which wasn't in the statement.
The statement, the statement. That's all we have. All we know is what
the statement "told" us.
Why
> is that a reasonable interpretation? Because that is the
> way people _do_ interpret the question! And whoever
> wrote the question presumably knows that people
> will interpret it that way - hence he would have written
> something else if he meant something else.
>
Now you're getting into what "whoever wrote the question presumably
knows". In my argument I never get into what the author presumes, or
what the author intended.
If the author intended one question, but asked another. Which question
should the answerer answer?
With my argument, I answer the question which was asked. That's why my
argument is true. That's why you can't refute it. All you can do is
bad mouth it.
Eldon Moritz
The key to the working model is that we are re iterating an event that
has already happened.
Suppose that the coins landed HT, and that the writer wrote "at least
one is a head" without prejudice.
Then, if we assume that the choice was made with prejudice, we have
changed the question. If we assume that, because "heads" were chose,
"heads" were pre chosen, we have changed the question.
Eldon
Eldon Moritz
> On 8 Apr 2003 09:28:55 -0700, elmo...@yahoo.com (Eldon Moritz) wrote:
>
> >David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<cq759vci010bl1aep...@4ax.com>...
> >> On 7 Apr 2003 16:04:46 -0700, elmo...@yahoo.com (Eldon Moritz) wrote:
> >>
> >> >David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<hj709vk79slr5tbv2...@4ax.com>...
> >> >> On 5 Apr 2003 12:10:05 -0800, elmo...@yahoo.com (Eldon Moritz) wrote:
> >> >>
> >> >> >We have been discussing coin flips in the "Counter-intuitive
> >> >> >mathematical results" thread, in this newsgroup.
> >> >> >
> >> >> >Our question:
> >> >> >Two coins were flipped and at least one is a tail. What are the
> >> >> >chances for two tails? or
> >> >> >Two coins were flipped and at least one is a head. What are the
> >> >> >chances for two heads?
> >> >> >
> >> >> >and should have the same answer. (we haven't agreed upon much else)
> [...]
> >> >> >
> >> >> >I have stated that model 1 seems to be the correct model for the
> >> >> >probability for two, given at least one. It gets correct answer 1/3
> >> >> >for that statement, and as such, defines what is meant by the term
> >> >> >"given at least one." It does not define what is meant by "at least
> >> >> >one is."
> >> >>
> >> >> In case anyone can't figure out what's meant here: He's insisting
> >> >> that the two questions
> >> >>
> >> >> chances for two tails?"
> >> >>
> >> >
> >> >Call this B'.
> >> >
> >> >> and
> >> >>
> >> >> "Two coins were flipped. Given that at least one is a tail, what
> >> >> are the chances for two tails?"
> >> >>
> >> >
> >> >Call this A'.
> >> >
> >> >> are different.
> >> >>
> >> >> Honest. I point this out expliictly for the benefit of readers
> >> >> who might try to figure out what his point is, and might
> >> >> have a hard time, because they'd never imagine that
> >> >> someone who spoke fluent English would think that
> >> >> the two were different questions.
> >> >>
> >> >
> >>
> >> They're the same.
> >>
> >> >If they are the
> >> >same, how could an English speaking person think they were different?
> >>
> >> This is a mystery to most of us. Could be the guy was an astonishingly
> >> dumb fuck, or it could be that somehow he acquired a serious
> >> misunderstanding of how the language is used in this sort of context.
> >> Or it could be a result of his obsession with the fact that He is
> >> Right about this and everyone Else is Wrong, leading him to
> >> misread simple English that he'd have no trouble reading correctly
> >> in a context unrelated to his obsession.
> >>
> >> All sorts of possibilities. But the fact that one English-speaking
> >> person thinks that A' and B' are different does not prove that
> >> they actually are.
> >>
> >How can you understand? You snipped all my points. I explained very
> >thoroughly how A' and B' are different.
>
> No, you explained why you _think_ that A' and B' are different.
>
Now we know. I'm answering the question as written. You're answering
the question you think the writer intended to write.
> >You completely ignored it.
>
> I've explained this before. It's not _possible_ to _prove_ that A' and
> B' are different, just as it's not possible to prove that they are the
> same. This is because they are expressed in English. The meaning
> of a string of English words is not something that can be worked
> out mathematically - it's a matter of convention. If you think that
> A' and B' mean different things it simply follows that you don't
> understand how the language is used. I ignore the details of
> your analysis of various "models" because they're simply
> irrelevant, being based on a misinterpretation of the statement.
>
It is silly argument to argue that we can't have stated questions in
English. We have them all the time.
> Such things _are_ defined by concensus. And I have not
> seen one person agree that A' and B' are different -
> everyone who's commented agrees they mean the same
> thing. Many people think it's very strange that anyone
> would think they're not the same.
>
You want mathematics by popular vote. I go by logical proof.
> (Um, if we know the meaning of a _clause_ P and a clause
> Q we can define the meaning of "P and Q" mathematicially.
> But points like whether the absence of the word "given" in
> B' implies that it means something different from what A'
> means is not the sort of thing that's suscpetible to proof
> and disproof.)
>
P ( 2 boys | at least one boy) is 1/3. That defines what it means to
say given at least one. You err when you take that to define "at
least one is". That's your error. That's why the truth is on my side.
That's why my models work, and yours don't. That's why I offer good
argument and you have to say "dumb fuck."
I answer the question as written. You answer the question you think
the writer intended to write.
We don't know what the writer intended. We only know what the writer
wrote.
> >Then you claim victory. This is one more case of you not understanding
> >my argument. I'm an asshole when I point out that I can't tell if you
> >can't, or won't.
> >
> >Arguing with you is like arguing with Sadaam Hussein. Note that I am
> >not saying that you are like Sadaam Hussein. He is an internationally
> >known scoundrel. You are an internationally known, well respected
> >educator from the great state of Oklahoma.
> >
> >You argue like Hussein. No matter what the facts are, you claim
> >victory. You ignore my argument, then claim victory. Sadaam seems to
> >have taken that same tactic to the ultimate extreme. Keep ignoring my
> >argument, you never will understand it.
> >
> >You have continually paraphrased my argument wrong. I think you don't
> >understand it. If you understood my argument, you would paraphrase it
> >right, then disagree.
> >
> >You can put your argument into logical terms. I can tell you where the
> >false statements are.
> >
> >I'll put my argument into logical terms. You can't tell me where the
> >false statements are, there aren't any, the truth is on my side.
>
> It's not that simple. At the _start_ of what you propose there's a
> step where the English words are _translated_ into "logical
> terms". And you're simply doing the translation incorrectly.
>
We can go step by step, and you can't win.
That's why you won't go step by step. You don't seem to want to know
the truth. The truth is on my side.
> [...]
> >
> >Don't claim victory without shooting down at least one point.
>
> You keep saying this. I _have_ shot down one point. Not
> with a mathematical proof, because the point in question
> is not the sort of thing that's susceptible to mathematical
> proof. But the point which has been shot down is the
> "point" that "at least one is a head" means something
> different from "given that at least one is a head".
>
Two coins were flipped. P ( 2h | at least one head) is 1/3. We both
agree on the math. That defines given at least one.
Suppose that a studen asked, what does that mean? What does it mean to
say given at least one? What would you say? What is that definition?
I can define it, or I'll get a definition from a mathematics
professor. It exists. I think you know, you just won't say. We'll go
point to point until we agree on that definition.
We may have to go to the English Lit department to get a definition
for "at least one is" but we can get it. We can work on examples, look
in dictionaries. All we have to do is find a good definition for "at
least one is".
Then we'll compare the two definitions. YOU CAN'T WIN this point.
> It's true that I haven't shot this point down by anything
> more compelling than just asserting that it's not so.
> But that's all that's available, because of the nature of
> the point in question. If "everyone" agrees that
> "... At least one is a head" means the same thing
> (in A' and B') as "... Given that at least one is a head"
> then they _do_ mean the same thing, because that's
> how language works. And in fact everyone _does_
> agree they mean the same thing.
>
So we can have a vote. Find out the true meaning of "at least one is".
We have a pretty good definition of given at least one. It's a
mathematical definition and it's exact. I think we can get a
concencous on this one.
The statement that everyone does agree that they mean the same thing
is another one of your inventions. You probably dreamed it.
> ("Everyone"? Well, I've talked to many many people
> about math for many years, and that's the way
> everyone I've ever spoken to uses the language.
> And there has been nobody here agreeing that
> they mean something different. On _this_ point,
> if you want to prove you're right, by _definition_
> you need to find _many_ people who _agree_
> you're right. You haven't done that, not even close.)
>
And while we're finding the definition for "at least one is", we can
also ponder this question.
If a writer writes one question, but intended another. Which question
should the answerer answer.
Eldon Moritz
> elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> > said "there was no asker." That's wrong. So far, you haven't
> > capitulated.We have a written question. It was written by someone, or
> > something.. Whatever asked the question, call whatever the asker.
>
> You have distorted, either deliberately or by misunderstanding,
> the discussion of the "asker".
>
> an author asking YOU, the reader something.
>
> The point of contention is whether a statement like
> "One of the coins is a heads" implies that the question
> author had to obtain this information by asking
> a question of SOMEBODY ELSE. Somebody who, in your
> interpretation, might have answered differently even
>
The point of contention:
It's a fact that the statement could not have been made without some
kind of inspection.
All we actually know about the inspection is that the asker had to
make one.
Details of the inspection are not important. What is important is what
the statement told us of the inspection. What information did we
receive? That's important.
Our statement told us that the asker found evidence of a head. There
is no evidence that the asker knew the outcome of the other coin.
If the asker only inspected one coin, the answer is definitely 1/2.
If the asker saw both coins, then made our statement, that's okay, so
long as information which would change the answer wasn't withheld. The
asker had to make a true statement. Information couldn't be withheld
which would make the statement false.
Eldon
David C. Ullrich
wrote:
>rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
>> elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
>> > I said something about what the asker of our question had to know. You
>> > said "there was no asker." That's wrong. So far, you haven't
>> > capitulated.We have a written question. It was written by someone, or
>> > something.. Whatever asked the question, call whatever the asker.
>>
>> You have distorted, either deliberately or by misunderstanding,
>> the discussion of the "asker".
>>
>> Of course somebody wrote the problem. That means there is
>> an author asking YOU, the reader something.
>>
>> The point of contention is whether a statement like
>> "One of the coins is a heads" implies that the question
>> author had to obtain this information by asking
>> a question of SOMEBODY ELSE. Somebody who, in your
>> interpretation, might have answered differently even
>> if there was in fact a heads among the flips.
>>
>The point of contention:
>
>It's a fact that the statement could not have been made without some
>kind of inspection.
No it's not. Look:
"x = 2. What is x + x?"
There. I just made the statement x = 2. And I did it without
any sort of inspection. qed.
>All we actually know about the inspection is that the asker had to
>make one.
>
>Details of the inspection are not important. What is important is what
>the statement told us of the inspection. What information did we
>receive? That's important.
>
>Our statement told us that the asker found evidence of a head. There
>is no evidence that the asker knew the outcome of the other coin.
>
>If the asker only inspected one coin, the answer is definitely 1/2.
>
>If the asker saw both coins, then made our statement, that's okay, so
>long as information which would change the answer wasn't withheld. The
>asker had to make a true statement. Information couldn't be withheld
>which would make the statement false.
>
>Eldon
******************
David C. Ullrich
David C. Ullrich
wrote:
So you think. In fact you're just showing that you don't understand
how simple English is used in stating mathematical problems.
>You're answering
>the question you think the writer intended to write.
>
>> >You completely ignored it.
>>
>> I've explained this before. It's not _possible_ to _prove_ that A' and
>> B' are different, just as it's not possible to prove that they are the
>> same. This is because they are expressed in English. The meaning
>> of a string of English words is not something that can be worked
>> out mathematically - it's a matter of convention. If you think that
>> A' and B' mean different things it simply follows that you don't
>> understand how the language is used. I ignore the details of
>> your analysis of various "models" because they're simply
>> irrelevant, being based on a misinterpretation of the statement.
>>
>It is silly argument to argue that we can't have stated questions in
>English. We have them all the time.
You continually give evidence that you have big problems
understanding simple English. The idea that the previous
paragraph says that we can't have stated questions in
English is the most recent example.
>> Such things _are_ defined by concensus. And I have not
>> seen one person agree that A' and B' are different -
>> everyone who's commented agrees they mean the same
>> thing. Many people think it's very strange that anyone
>> would think they're not the same.
>>
>You want mathematics by popular vote. I go by logical proof.
And here's another. Of course mathematics is based on
logical proof, not popular vote - I haven't hinted otherwise.
What _is_ based on popular vote is how English statements
are translated into mathematical concepts.
[...]
>
>We may have to go to the English Lit department to get a definition
>for "at least one is" but we can get it. We can work on examples, look
>in dictionaries. All we have to do is find a good definition for "at
>least one is".
>
>Then we'll compare the two definitions. YOU CAN'T WIN this point.
Depends on what WIN means. It's certainly true that I can't
get you to see where your error is. But in fact I've already
won - _nobody_ agrees that A' and B' say different things;
_everyone_ finds the notion that they say different things
to be extremely wacky [note slight correction to this
statement below].
The Truth is on Your Side. Right. If we were talking about
a mathematical question it could in fact be that you were
right and everyone else was wrong - you have a new proof
that 2 + 2 = 5, it _could_ be that you're right and nobody
ever noticed that everyone's been wrong for millenia
thinking that 2 + 2 = 4. Unlikely, but it could be.
But we're not talking about a mathematical question.
We're talking about what a certain sequence of
English words _means_. In such a context, if
everyone thinks you're wrong then you _are_
wrong. By definition.
But don't let that persuade you. If you did you'd
become much less entertaining. (Like yesterday
playing backgammon, I was giving the other
guy an update on the latest sci.math wackiness.
I said there was this guy insisting that A' and B'
meant different things. He couldn't hear any
difference - I had to write them down. He
couldn't believe that there was someone in
the universe who thought they were different.)
Slight correction: The statement that nobody
thinks that A' and B' are different is a slight
exaggeration - yesterday we saw Buhr explain
what he thought the difference was. But he
got 1/3 for both of them - the difference was
in details of the mathematical model that
have _no_ effect on the answer to the
question, taking one as a statement about
conditional probabilities and the other as
a statement about a restriction on the
sample space. Since one calculates
conditional probabilities _by_ restricting
things to a subset of the original sample
space this is very close to being a
difference that makes no difference -
the two "models" he obtained from
A' and B' are equivalent, as far as the
answer to the question goes.
>> It's true that I haven't shot this point down by anything
>> more compelling than just asserting that it's not so.
>> But that's all that's available, because of the nature of
>> the point in question. If "everyone" agrees that
>> "... At least one is a head" means the same thing
>> (in A' and B') as "... Given that at least one is a head"
>> then they _do_ mean the same thing, because that's
>> how language works. And in fact everyone _does_
>> agree they mean the same thing.
>>
>So we can have a vote.
We've been doing that. Haven't you noticed? Nobody
but you thinks that A' and B' are different!
>Find out the true meaning of "at least one is".
>We have a pretty good definition of given at least one. It's a
>mathematical definition and it's exact. I think we can get a
>concencous on this one.
>
>The statement that everyone does agree that they mean the same thing
>is another one of your inventions. You probably dreamed it.
No doubt. Remind me who has agreed that they're different.
(Um, remind me who has agreed that they're different enough
that the answers are different.) I recall many posts on the
subject from many people, but I don't recall the ones you
seem to be alluding to here.
>> ("Everyone"? Well, I've talked to many many people
>> about math for many years, and that's the way
>> everyone I've ever spoken to uses the language.
>> And there has been nobody here agreeing that
>> they mean something different. On _this_ point,
>> if you want to prove you're right, by _definition_
>> you need to find _many_ people who _agree_
>> you're right. You haven't done that, not even close.)
>>
>And while we're finding the definition for "at least one is", we can
>also ponder this question.
>
>If a writer writes one question, but intended another. Which question
>should the answerer answer.
>
>Eldon
>
>
>>
>>
>> ******************
>>
>> David C. Ullrich
******************
David C. Ullrich
David C. Ullrich
wrote:
Exactly. This would be different. (And this _different_ question is
the sort of question that Eldon is actually answering. He continually
denies this, but when you look at his analyses you see it's so.)
>and (s)he answers "heads". Now
>the probability of two heads is 1/2 under the most reasonable
>interpretations (which involve assuming the person we're asking has no
>heads/tails bias).
>
>Michael
******************
David C. Ullrich
Eldon Moritz
> > rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
> > > elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> > > > I said something about what the asker of our question had to know. You
> > > > said "there was no asker." That's wrong. So far, you haven't
> > > > capitulated.We have a written question. It was written by someone, or
> > > > something.. Whatever asked the question, call whatever the asker.
> > >
> > > You have distorted, either deliberately or by misunderstanding,
> > > the discussion of the "asker".
> > >
> > thing is that I know nothing except what I learned from the statement.
> > The asker told me via the statement.
>
> See? Now you're just lying, because in your very next paragraph
> you introduce the third party again. The bone of contention
> is the existence of this third party.
>
I do not introduce a third party in my next statement. Eldon Moritz
> > > The point of contention is whether a statement like
> > > "One of the coins is a heads" implies that the question
> > > author had to obtain this information by asking
> > > a question of SOMEBODY ELSE. Somebody who, in your
> > > interpretation, might have answered differently even
> > > if there was in fact a heads among the flips.
> > >
> > We know that our statement couldn't have been made without some kind
> > of investigation.
>
This was my next statement. There is no mention of a third party here.
Eldon Moritz
> Really? Then why do you accept "given that one of the coins
> is a heads" without the investigation scenario?
>
I accept "given" as a special mathematical word. It is defined by
mathematical definition. This leaves room for ambiguity if it is used
in a non mathematical forum. When this problem was introduced in
Parade Magazine, a family magazine, it did not contain the word
"given". If it had, I would have demurred.
> If I'm writing a test, I'm creating the universe. I don't have
> to investigate to know the facts in this universe. That's
> elementary. If you are writing a test problem about
> coin flips, you don't have to go flip some coins to write
> it. You can make up hypothetical situations all you want in
> your head, and you can establish the facts that you want
> the test-taker to be aware of.
>
Hypotheticals are nice. It is easy to move the subjects around. The
rules of fair play still apply. Just be careful that the 'facts' you
establish are the facts which you want established.
Ullrich and I have agreed that he was answering the question which he
thought the author intended.
I have alway answered the question as written.
That's two different questions.
> > Some way, some how, the writer of the question knew
> > that there were two coins.
>
> It's his story. He MADE IT UP. You deny an author's ability
> to know the truth of HIS OWN STORY?
>
Certainly not. Sometime, as Ullrich and I have agreed, the author can
THINK one story, and WRITE another. Do you deny that this can happen?
> > We also know that the writer found out the
> > outcome of one coin. In our question there is no evidence that the
> > writer knew the outcome of both coins.
>
> Except when the word "given" is present. Then he regains
> his ability to know the facts of the story he is writing.
>
That's true. A' was a crudely written question trying to write a
question for the mathematical definition. P ( 2 | at least one) = 1/3.
A' has definite answer 1/3 depending on whether or not it is accepted
as this question.
B', or "Two coins were flipped, at least one is a head. What are the
chances for two heads?" is a stated question. As written it has
correct answer 1/2.
If you answer B' as the question which Ullrich thinks the author
intended. Then it has answer 1/3.
> > > For some reason you accept that the question author
> > > can have test-givers omniscience when saying "given
> > > that one of the coins is a heads" but when the
> > > author leaves out the word "given" he loses his
> > > omniscience and must rely on polls and unreliable
> > > third parties.
> > >
> > I say that the mathematical statement the probability for two, given
> > at least one confuses folks.
>
> It shouldn't. The part after the bar in a conditional probability
> defines a subset of the universe. Define that subset correctly
> and you can figure out which fraction has the event "two heads".
Yes, Mathematically: the probability for two, given at least one is
1/3.
That is a mathematical definition. It defines what it means to say
"given", or it defines what it means to say "given at least one."
Mathematically..
It confuses many mathematicians into believing that it defines "at
least one is." It doesn't.
> > I'm not arguing "given". I'm arguing our question. Don't let "given"
> > get in the way. I have asked what it means to say given. No one seems
> > to want to answer. Our question doesn't have given in it, so it's
> > moot. That's my argument.
>
> How can you think it's "moot" if it's absence changes the answer,
> in your opinion? You're lying again. Obviously you think it
> matters.
>
In arguing our question, "Two coins were flipped and at least one is a
head. What are the chances for two heads?" it's moot what 'given'
means because it isn't in our question.
In arguing A', it isn't moot, because it does change the question.
Change the definition of 'given' and you change the answer for A'.
Change it to hell and back, it doesn't affect B'.
I wasn't lying, I was misunderstood.
> > Don't say our argument has the answer 1/3, because of what "given"
> > means. I didn't say the writer had to rely on third parties, when
> > given wasn't in the question.
>
> You said the writer had to do an investigation, as if writers
> are incapable of defining hypothetical situations.
>
Define a hypothetical situation, but the rules of the game still
apply. The words of the question still prevail. Can you agree with
Ullrich and me that the author thought one question, then asked
another.
All we have to go by is the words in the question, the statement.
> > I said that the writer had to write a
> > true statement. To write a true statement the writer had to have had
> > some kind of inspection.
>
> Or, as a writer, he could have MADE IT UP.
>
That's okay with me. We still have to go by WHAT THE STATEMENT SAID.
If the author thought one question, then asked another. Which
statement should the answerer answer?
Eldon
> - Randy
Randy Poe
wrote:
>rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
>> See? Now you're just lying, because in your very next paragraph
>> you introduce the third party again. The bone of contention
>> is the existence of this third party.
>>
>I do not introduce a third party in my next statement. Eldon Moritz
As soon as you get into "the person speaking had to OBTAIN THIS
INFORMATION somehow", you start hypothesizing about a third party the
writer needed to ask.
The writer is a WRITER. He's writing a STORY. He KNOWS the truth
because he's MAKING IT UP. That's what writers do. That's why we
interpret what the writer says, i.e. when he says "one coin is a
heads" we take him at his word. We don't interpolate extra story lines
that we think the writer intended.
Stick to what the writer wrote. When the writer wrote "one coin is a
heads" assume the writer knows that one coin is a heads. Assume this
was not an EXPERIMENT, but a universe made up by the writer (he's the
WRITER, remember?) and he has omniscient knowledge of HIS OWN G-D
UNIVERSE.
- Randy
David C. Ullrich
wrote:
[...]
>
>Ullrich and I have agreed that he was answering the question which he
>thought the author intended.
Yes. In other words, answering the question as written.
>I have alway answered the question as written.
Uh, no.
[...]
>>
>That's okay with me. We still have to go by WHAT THE STATEMENT SAID.
Nobody is disputing that. The question is whether you are correctly
interpreting what the statement says. (Hint: the answer is no.)
>If the author thought one question, then asked another. Which
>statement should the answerer answer?
>
>Eldon
>
>> - Randy
******************
David C. Ullrich
Eldon Moritz
> >All we actually know about the inspection is that the asker had to
> >make one.
> >
> >Details of the inspection are not important. What is important is what
> >the statement told us of the inspection. What information did we
> >receive? That's important.
> >
> >Our statement told us that the asker found evidence of a head. There
> >is no evidence that the asker knew the outcome of the other coin.
> >
> >If the asker only inspected one coin, the answer is definitely 1/2.
> >
> >If the asker saw both coins, then made our statement, that's okay, so
> >long as information which would change the answer wasn't withheld. The
> >asker had to make a true statement. Information couldn't be withheld
> >which would make the statement false.
> >
> >Eldon
>
>
> ******************
>
David C. Ullrich
wrote:
Very compelling response.
>> >All we actually know about the inspection is that the asker had to
>> >make one.
>> >
>> >Details of the inspection are not important. What is important is what
>> >the statement told us of the inspection. What information did we
>> >receive? That's important.
>> >
>> >Our statement told us that the asker found evidence of a head. There
>> >is no evidence that the asker knew the outcome of the other coin.
>> >
>> >If the asker only inspected one coin, the answer is definitely 1/2.
>> >
>> >If the asker saw both coins, then made our statement, that's okay, so
>> >long as information which would change the answer wasn't withheld. The
>> >asker had to make a true statement. Information couldn't be withheld
>> >which would make the statement false.
>> >
>> >Eldon
>>
>>
>> ******************
>>
>> David C."Dumfuck" Ullrich
Huh. Now you're modifying _quotes_ - _explicitly_ stating that
people said things that you _know_ they did not say.
Lying, in other words.
Calling me a dumb fuck the first time turned out to be a mistake.
Saying I said something I didn't say, that you simply made up,
may turn out to have been bad tactics as well.
******************
David C. Ullrich
Eldon Moritz
> elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> > rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
> > > elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> > > > Two coins flip four ways. To flip two coins three ways, the
> > > > heads/tails decision must be made prior to the flip,
> > >
> > > Who said anything about "flipping two coins three ways"?
> > >
This email articulates the difference between what I'm saying, and
what you think I'm saying.
For this discussion, define before and after the flip as before and
after the look. Prior to the look, you may reflip, it changes nothing.
After the look, we have a flip. After the look, we are after the flip.
> > When you flip two coins with three equally likely outcomes, you have
> > flipped two coins, three ways. Flip two coins, you get fourths. To get
> > a correct answer of 1/3, you must change this to thirds.
>
> By selecting a subset of flips after the fact.
>
Not true. Somewhere along the way, the heads/tails decision must be
made. To get thirds, you can select 'heads', or you can select
'tails'. You can get a subset of three, either way. To get this
subset, and for them to be three equally likely, you must select,
prior to the flip.
Bill and Joe always bet, Bill always bets for two of a kind. Joe
always bets for one of each. When you look, their bet is on.
Flip two coins and look at them. You are left with one outcome. It is
one of four.
> > When two coins are flipped, there are two alike on each end. HH, HT,
> > TH, TT. In other words, flip two coins and you get fourths. To get
> > thirds, you must eliminate one, prior to the flip.
>
> Or after.
Not true. I made a true statement. You made a false statement. If you
flip two coins, then look, and see one of three. You had a 'gender' in
mind, prior to the flip. Bill has bet for two of a kind, Joe has bet
for one of each. If you flip and look, Bill and Joe's bet is on. On
this flip.
They have a fifty fifty bet, no matter what you say. For them to have
a two for one bet, you MUST eliminate one of the four, prior to the
look. Try it.
>
> > > Conditional probability means sampling from a subset. In this
> > > case, it means sampling from the subset of two-coin flips that
> > > contain a heads. No pre-decision is needed. The decision is
> > > made after the fact.
> > >
> > Here we differ. In this case it means sampling from the subset of two
> > coin flips about which the "at least one is a heads" statement was
> > made.
>
> Which can be made after, by looking at both flips. Your
> error is in assuming that I wouldn't tell you one is a heads
> after I look at both. That I might always choose to answer
> something differently, even if I tell you EXPLICITLY I
> am never going to say "one is a heads". I tell outright
> that I am going to look at the flips and make either the
> statement "one is a heads" or "there are no heads", and
> no other statement.
>
So long as you make this 'for the heads' decision, prior to the look,
prior to the flip. The kicker is did you decide 'heads' prior to the
look? In the above example, you did. Then, the statement is true or
not, depending upon whether or not you correctly told our bettors what
you did. All they get is the statement. All they have to do to make
the proper bet, is to answer the question correctly.
> > You are sampling from the subset of which our statement would have
> > been true.
>
> Right. Now, of which subset of flips is the statement "one
> is a heads" true. Not "did a hypothetical and mischievous
> observer happen to say one is a heads" but "one is a heads".
>
We have a statement about a coin flip. For it to be a third, instead
of a fourth, someone, or something, had to have selected heads, prior
to the flip. Prior to the inspection. Prior to the 'look'.
> Look at this flip: HT. Is the statement "one is a heads" true?
> If so, I include it in my subset. Always.
>
Yes, and you have decided this, prior to the flip.
> Look at this flip: TH. Is the statement "one is a heads" true?
> If so, I include it in my subset. Always.
>
Yes.
> Look at this flip: HH. Is the statement "one is a heads" true?
> If so, I include it in my subset. Always.
>
Yes, and you have made this decision, prior to the flip.
> Look at this flip: TT. Is the statement "one is a heads" true?
> If not, I do not include it in my subset. Ever.
>
This gets a flip of thirds. Three equally likely.
> As you say, the subset is that subset of flips, after the
> fact, for which the STATEMENT is true.
>
Hopefully, this illustrates what I meant. I have said, over and over,
to get thirds for heads, you must reflip, or demur on TT. You did
this, it works.
How do you explain it to Bill and Joe? All they get is what you say in
'the statement.'
> > > Here's another conditional probability scenario for you: I
> > > do a random poll of people on the street about their shopping
> > > habits. After I get home, my client wants to know what
> > > proportion of people who shop at Target also shop at Walmart.
> > > That is, "What is the probability X shops at Walmart given
> > > X shops at Target?". In your view as stated above, I would
> > > have had to pre-determined my sample and shot all the non-Target
> > > people so they wouldn't contaminate the sample.
> > >
> > Different analogy entirely.
>
> Nope. Same concept: Proportion of a subset, which can be
> chosen after the fact.
>
> > > On our planet, since a conditional probability represents
> > > a probability on a subset, all you have to do is select that subset
> > > and you can do that after your experiment.
> > >
> > Yes, you can, but you have to determine the proper subset.
> >
> > ***You cannot, on this planet, change "fourths" to "thirds", after you
> > look at the toss.*** That's a challenge. If you can do it, I'll
> > capitulate.
>
> Subset. Subset. Subset. What is so hard about taking a subset?
> Think about my polling example. I can indeed change from
> "proportion of all shoppers who shop at Walmart" to "proportion
> of Target shoppers who shop at Walmart" and I can do it
> after the fact. It's EASY!!!
>
We're talking about one toss, one statement. That's what we have.
> I do it this way: I'm keeping a tally sheet with two columns.
> One is total number of two-flip tosses that include a heads,
> the other is flips with two heads. Every time heads comes
> up at least once, I make a mark in column A. When it comes
> up twice, I make a mark in column B. When two tails comes
> up, I make no mark.
>
> At the end of the experiment, I divide the number of marks
> in column B by the number of marks in column A.
>
> I can even write a simple computer program designed to do
> this, and it doesn't have to prejudice the flips beforehand.
> It just has to do this (pseudo-code):
>
> flip = random_coin_toss();
> if (flip == 'HT') {colA++;}
> else if (flip == 'HH') {colA++; colB++}
> else {} /* Two tails: do nothing */
>
This says that, you have decided to gather the subset of flips, about
which the heads statement is true.
> Does that bother you? How about this?
>
> flip = random_coin_toss();
> if (flip == 'HT') {colA++; total++}
> else if (flip == 'HH') {colA++; colB++; total++}
> else {total++;}
>
> There. Now I'm counting every toss that occurs (in
> "total"), throwing out nothing. But colA is only
> about 3/4 of total, and colB is about 1/3 of colA,
> or 1/4 of total.
>
> I chose my subset after the flip, just as I said. Why is
> that such a hard concept?
>
I know the ratios which should come up in a series of two coin flips.
You can make a subset for tails, and one for heads. What you can't do,
and what I've said that you can't do, is change the odds on Bill and
Joe's bet, after you look at the flip.
I repeat my challenge. Flip two coins, without prejudice. Then look.
Then make a statement about that flip which changes the bettors odds.
> - Randy
What I'm trying to tell you is that 'the key' to this question, is
whether the coins were flipped with, or without prejudice.
To get one of three, you must 'flip' with prejudice.
Flip, and then look, without prejudice. It's one of four.
Eldon
David C. Ullrich
wrote:
>rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03041...@posting.google.com>...
>> elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
>> > rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
>> > > elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
>> > > > Two coins flip four ways. To flip two coins three ways, the
>> > > > heads/tails decision must be made prior to the flip,
>> > >
>> > > Who said anything about "flipping two coins three ways"?
>> > >
>
>This email articulates the difference between what I'm saying, and
>what you think I'm saying.
>
>What I'm actually saying is that I get strange answers to these
>questions because I simply don't understand the meaning of
>perfectly standard English constructions.
Ah. Yes, we have agreement on that.
[...]
Kevin Buhr
> >
> >[[For the record, I think both interpretations are incorrect. I think
> >the correct mathematical interpretation of this poorly posed question
> >is that the sample space is Omega'={HT,TH,TT}, the appropriate
> >probability on it has to be assumed but a good convention *might be*
> >to take it to be P_1(omega) = P_0(omega|Omega'), and I get the same
> >answers as you do,
>
> Hence I don't see the point to the distinction.
I didn't explain it that well. I didn't mean to suggest (or I regret
having suggested) that it was terrifically important whether you start
with Omega and assume questions about probabilities are questions
about conditional probabilities P_0(.|Omega') or start with Omega' and
use a P_1 that happens to be equal to that conditional probability.
The important distinction is that the better-posed question relies on
a very standard convention to establish the probability space, and the
definition of conditional probability permits no argument; in
contrast, the poorly posed question requires us to invent a convention
for the probability space. A good one *might be* the one I suggested,
but I'm by no means convinced. For example, consider the following
two scenarios:
1. You are talking to a woman. She mentions she has two children,
but doesn't say anything else about them. Later in the conversation,
you ask her, "do you have a son?" She says, "yes", and seems on the
verge of adding either "two of them" or "he's a handful compared to
his sister" or who knows what when you are unfortunately interrupted.
2. You are talking to a woman. She mentions that she has two
children. Then, she says, "my children are always getting into
trouble, especially Gregory."
If you're a psychologist, helping the woman deal with the many
stresses of child-rearing, you might reasonably jot down in your
notebook "two children, at least one boy" in either scenario, and no
one would fault you for not accurately documenting the situation.
If you're a *bad* psychologist, and later break confidentiality by
sharing this information with a mathematically inclined friend, you
could reasonable say, "I happen to know this woman with two children,
at least one of which is a boy. Say, what are the chances that she
has two boys?"
If your mathematician friend tries to answer, he is reading something
significant into the problem. He has no a priori reason to believe
that the first type of scenario is more or less probable than the
second type.
In fact, in most of the realistic scenarios I can imagine where you'd
obtain the information that at least one child was a boy and only that
information, the answer *is* 1/2. This is because people who suffer
from no mental impairment won't literally answer the question "do you
have a boy?" with "yes, at least one". I had to invent an
interruption in scenario number 1 above to make it seem plausible. On
the other hand, people *will* let drop the gender of one child or the
other on occasion.
And here, the poser of the question isn't, presumably, asking for "the
conditional probability of two boys given one boy". The poser wants a
genuine probability in light of the available information---the
probability you'd use to assess your chances of winning a bet or the
limiting proportion of times there'd actually be two boys if this
random experiment could be repeated many times, even if the poser
wouldn't put it in exactly those terms.
Eldon's complaint when he says "many mathematicians don't know there's
a difference" is, I think, that math types tend to have a hard-wired
part of their brains where---if a problem includes some information
and asks for the chances of this or the probability of that---we must
write down "P(A|B)=" and continue from there. In fact, in most
real-world probability problems, the context in which the information
is delivered is at least as important as the information itself, and
the "difficulties" we have with certain toy problems like the two-sons
problem are getting at a deeper truth.
I'd go even further. It would be wrong even for the hypothetical
mathematician above to say, "from the information you've given me, I
would say 1/3." If the mathematician is offering an (unconditional)
probability, then he must have a specific random experiment (i.e., a
specific probability space) in mind where the event "two boys" has
probability 1/3. What is it? It's not difficult to figure out what
the mathematician is imagining, but without prior probabilities for
scenarios 1 and 2 and all the other scenarios, it seems unjustified.
> Yes, it's certainly true that the "at least one is a head" is not
> worded very well - in a formal mathematical context I'd say it
> needed to be restated. But this sort of wording is very common
> in informal descriptions of puzzles - when we read "One and a
> half chickens can lay one and a half eggs in one and a half
> days..." we don't assume that the writer is actually asserting
> something that's supposed to be _true_, we understand that
> he's giving us hypotheses, even though the gramattical
> markers indicating that they are hypotheses have been
> omitted.
>
> In particular we _don't_ start analyzing the problem on
> the basis of how we obtained the information; on the
> basis of who said this and why...
I hope I've made a reasonable case for why, unlike in the chicken
problem, we *should* start anaylzing the two-sons problem (or the
Monty Hall problem or many other similar problems) on the basis of how
we obtained the information.
Someone who nitpicks the chicken problem is being annoying. Someone
who nitpicks the two-sons problem (or the Monty Hall problem) is
often, I feel, getting to its heart. The real difference is that the
information in the chicken problem is obviously attempting to describe
a constant chicken egg-laying rate. However, the information in the
two-son problem is just describing the occurrence of an event which we
are then expected to decompose into subevents whose probability we
must estimate! And the *way* we decompose this event is fundamental
to the solution of the problem---that's its whole point.
It's rather as if the chicken problem said "a duck and half a chicken
can lay two eggs in an hour". It certainly feels as if critical
information has been left out of the problem when we're then asked how
fast ducks lay eggs. A half a chicken must be dead, so a duck lays
two eggs an hour? Ducks and chickens lay eggs at the same rate,
so...? The two eggs mentioned must be a chicken egg and a duck egg,
so...?
I do see your point though. The fact that the 1/3 answer requires an
implausible backstory to make it "correct" in the sense I'm describing
is balanced on the other hand by the fact that, in the absence of any
mention of a backstory, 1/3 seems like the most direct answer given
the information presented.
> > [ some weird observational process I postulated ]
> As I've pointed out many times, _if_ we take the question that
> way then there's no way to give any sort of answer - how did we
> determine that the above is the "reasonable" assumption?
I think half the point of these kinds of questions is knowing when and
in what way the problem statement is incomplete. I think Eldon is
wrong in thinking 1/2 is The Right Answer. I think he is right in
realizing that the problem probably *isn't* asking for the conditional
probability mathematicians usually think it's asking for, and so 1/3
isn't The Right Answer either. This is true *even* if we make
allowances for the informal statement of the problem and its intended
meaning.
> It seems
> to me that he's simply misunderstanding the meaning of a
> certain informal English construction - I haven't given any
> "mathematical" arguments that it is what I say it is, because
> that's not a mathematical question.
I guess I feel that that informal English constructions are getting at
something more subtle than the formal request for a conditional
probability. I think mathematicians may be answering a slightly
different question than the one the poser intended.
Technically, the mathematicians' answer might be "right", but it's not
very right. It papers over a big hole in the question that the poser
should be told about.
--
Kevin <bu...@telus.net>
Eldon Moritz
>
> Eldon's complaint when he says "many mathematicians don't know there's
> a difference" is, I think, that math types tend to have a hard-wired
> part of their brains where---if a problem includes some information
> and asks for the chances of this or the probability of that---we must
> write down "P(A|B)=" and continue from there. In fact, in most
> real-world probability problems, the context in which the information
> is delivered is at least as important as the information itself, and
> the "difficulties" we have with certain toy problems like the two-sons
> problem are getting at a deeper truth.
>
If the mathematician doesn't watch out he'll outsmart himself, and
never get the correct answer to my question.
Consider the following 24 coin tosses. Bill and Joe are our bettors.
All they have to do to make the proper bet is answer the question
correctly. All they hear is the statement, and the question. All they
know about the toss, they learned from the statement.
1. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
2. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
3. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
4. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
5. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
6. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
7. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
8. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
9. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
10. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
11. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
12. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
13. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
14. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
15. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
16. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
17. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
18. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
19. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
20. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
21. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
22. Two coins were tossed and at least one is a head. What are the
chances for two heads? Bill bets for two heads, Joe bets for one of
each.
23. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
24. Two coins were tossed and at least one is a tail. What are the
chances for two tails? Bill bets for two tails, Joe bets for one of
each.
So long as statements to tosses are on a one to one ratio, Bill and
Joe have an even money bet. To get a two to one bet, the tosses to
statements must go 4 to 3. Some way it must be communicated through
the statement.
As a group therefore, it's easy to see that these statements all have
correct answer 1/2.
Individually, they should have the same answer. For instance take
number 19.
19. Two coins were tossed and at least one is a head. What are the
chances for two heads?
If, because it says, "at least one is", you immediatly go to
"P(A|B)=", how will you get the correct answer? If you hard headedly
say that B is at least one, therefore we were given at least one. You
will always get answer 1/3.
That's my point, my question exists, it has correct answer 1/2. You
have to do something different from what you are doing, or you can
never correctly answer my question.
Eldon
Denis Feldmann
published twice
Your analysis is really brillant. Thank you for having so clearly (and so
politely :-)) laid the real issues at stake here. I, in particular, had not
realized that the true reason of those difficulties is the ambiguity of the
real-life scenarios giving rise to the initial riddle, where mathematicians
like myself only see a clear-cut case of Bayesian analysis
ma...@mimosa.csv.warwick.ac.uk
Kevin Buhr <bu...@telus.net> writes:
Yes, I agree with everything you say below. I think that Eldon has been making
some valid points, and he does not really deserve to be treated with the
derision that he has received from some of the people who do not agree
with him. But on the other hand, his insistence that there is a unique
correct interpretation of the question as posed, and that the unique correct
answer is 1/2 really is absurd. The point about this problem is that
virtually all of the arguments put forward for either 1/2 or 1/3 are
possible interpretations of the problem. Some might be more plausible
than others, but it makes no sense to say that one is absolutely right
and another is absolutely wrong.
Some time ago somebody (forget who) suggested the following plausible
scenario for a 1/3 answer. You are the principal of a local boys school,
and on the lookout for new recruits. A newcomer to the area remarks mentions
that she has two young children. "Any boys?" you ask her. "Why yes!"
she replies.
Derek Holt.
David C. Ullrich
>David C. Ullrich <ull...@math.okstate.edu> writes:
>> >
>> >[[For the record, I think both interpretations are incorrect. I think
>> >the correct mathematical interpretation of this poorly posed question
>> >is that the sample space is Omega'={HT,TH,TT}, the appropriate
>> >probability on it has to be assumed but a good convention *might be*
>> >to take it to be P_1(omega) = P_0(omega|Omega'), and I get the same
>> >answers as you do,
>>
>> Hence I don't see the point to the distinction.
>
>I didn't explain it that well. I didn't mean to suggest (or I regret
>having suggested) that it was terrifically important whether you start
>with Omega and assume questions about probabilities are questions
>about conditional probabilities P_0(.|Omega') or start with Omega' and
>use a P_1 that happens to be equal to that conditional probability.
>
>The important distinction is that the better-posed question relies on
>a very standard convention to establish the probability space, and the
>definition of conditional probability permits no argument; in
>contrast, the poorly posed question requires us to invent a convention
>for the probability space.
It doesn't _require_ that. The poorly posed version is indeed poorly
posed, but it seems to me that _since_ it's poorly posed the first
thing we need to do is decide exactly what it means. And since
it's quite standard in informal statements of puzzles to write
hypotheses as assertions instead of hypotheses, it seems to
me the reasonable choice is to interpret the "At least one is
a head" to mean simply "given that at least one is a head...".
>A good one *might be* the one I suggested,
>but I'm by no means convinced. For example, consider the following
>two scenarios:
>
>1. You are talking to a woman. She mentions she has two children,
>but doesn't say anything else about them. Later in the conversation,
>you ask her, "do you have a son?" She says, "yes", and seems on the
>verge of adding either "two of them" or "he's a handful compared to
>his sister" or who knows what when you are unfortunately interrupted.
>
>2. You are talking to a woman. She mentions that she has two
>children. Then, she says, "my children are always getting into
>trouble, especially Gregory."
>
>If you're a psychologist, helping the woman deal with the many
>stresses of child-rearing, you might reasonably jot down in your
>notebook "two children, at least one boy" in either scenario, and no
>one would fault you for not accurately documenting the situation.
Seems to me that in both of these you're inserting much
more than is there.
Especially in (2). Given that she happened to _say_ something
that implied that at least one is a boy I don't see how we can
reasonably deduce _anything_ about the probabilities. We'd
need to know the answer to various poorly defined empirical
questions, regarding who says what under what circumstances.
_When_ we analyze the problem using this sort of "scenario"
it seems to me that there's simply no answer available -
one can invent a scenario that will give whatever answer
one wishes. Seems to me the only way one can give an
answer to the question is to take the "at least one is a
boy" as "given that at least one is a boy". That's assuming
exactly what's given in the problem, not adding other
factors.
You can hope that. Doesn't seem reasonable to me.
If the problem _were_ describing the result of some process
whereby we obtained the information, then yes indeed we'd
need to know much more about _how_ we obtained this
information - that's so true that if the problem _were_ about
the result of an experiment then the answer is clearly just
that there's not enough information given to answer the
question. Hence the problem must not be about the
result of some experiment.
Consider the questions Randy Poe asked once, that Eldon
decided to ignore instead of answer. When you read this:
"A chicken and a half can lay an egg and a half in a day
and a half. How many eggs can a chicken lay in a day?"
do you have any problem realizing that the first sentence
actually begins with the word "given"? Presumably not -
if the first sentence is meant literally, as an assertion, then
it raises the question of _which_ chicken and a half can
lay[etc]. And then since there's no relation given between
the 1.5 chickens about which something is asserted in
the first sentence and the chicken that's referred to in
the question constituting the second sentence, there's
no answer to the question.
But that's simply not what the question means.
> I think mathematicians may be answering a slightly
>different question than the one the poser intended.
>
>Technically, the mathematicians' answer might be "right", but it's not
>very right. It papers over a big hole in the question that the poser
>should be told about.
******************
David C. Ullrich
David C. Ullrich
<denis.f...@wanadoo.fr> wrote:
>Answer below, but I am mostly quoting , as I feel this is worth being
>published twice
>
>Kevin Buhr wrote:
>> David C. Ullrich <ull...@math.okstate.edu> writes:
>>>>
[...]
>
>Your analysis is really brillant. Thank you for having so clearly (and so
>politely :-)) laid the real issues at stake here. I, in particular, had not
>realized that the true reason of those difficulties is the ambiguity of the
>real-life scenarios giving rise to the initial riddle, where mathematicians
>like myself only see a clear-cut case of Bayesian analysis
What makes you think that the question is referring to some
unspecified real-life scenario in the first place?
******************
David C. Ullrich
David C. Ullrich
ma...@mimosa.csv.warwick.ac.uk () wrote:
>In article <87wuhz7...@saurus.asaurus.invalid>,
> Kevin Buhr <bu...@telus.net> writes:
>
>Yes, I agree with everything you say below. I think that Eldon has been making
>some valid points, and he does not really deserve to be treated with the
>derision that he has received from some of the people who do not agree
>with him.
"Derision"?
Have you read the entire threads? He was complaining about people
calling him stupid for some time - I looked and I couldn't see
_anywhere_ where people _were_ calling him stupid. Yes, there
has been a good deal of derision, but (as with Harris) it looks to
me like it started after he'd been behaving like an ass for some time.
For example, stating that people with degrees should be able to
see he's right. Stating that the reason I don't agree with him is
that I can't admit I'm wrong (that was stated _several_ times, and
it's about as insulting as anything I can imagine). Calling people
dumb fucks because they disagree (even though they've explained
clearly and politely _why_ they disagree.)
Derision indeed.
>But on the other hand, his insistence that there is a unique
>correct interpretation of the question as posed, and that the unique correct
>answer is 1/2 really is absurd.
That's true (except that of course I'd delete the "But on the other
hand".) And it's also exactly the sort of statement about what he's
said that he insisted on taking personally, as insults on his
character and intelligence, which then led to him making
_actual_ insults on people's character and intelligence.
******************
David C. Ullrich
David C. Ullrich
<denis.f...@wanadoo.fr> wrote:
>Answer below, but I am mostly quoting , as I feel this is worth being
>published twice
>
[...]
>
>Your analysis is really brillant. Thank you for having so clearly (and so
>politely :-)) laid the real issues at stake here.
Three questions about that:
Was it polite for him to call me a dumb fuck?
Was it polite for him to state repeatedly that the only reason I
didn't agree with him was that I can't admit it when I'm wrong?
Can you find something impolite that I said to him _before_ he
said those things to me?
>I, in particular, had not
>realized that the true reason of those difficulties is the ambiguity of the
>real-life scenarios giving rise to the initial riddle, where mathematicians
>like myself only see a clear-cut case of Bayesian analysis
>
>
******************
David C. Ullrich
Denis Feldmann
Mostly because of the Monty Hall problem. There, what people were "really"
asking was "is it smart to switch?", which cannot be decided without knowing
the scenario. I agree of course with you in reading the question (or the
questions with or without "given" ; I cannot see the difference) as being a
Bayesian question (and a trick question at that). But you must admit there
is room for scenarios interpretation, and in fact this is the reason I never
ask this kind of question, but questions with explicit scenarios. Anyway,
the whole point (and here, again, I agree with you) is that all this is not
a matter of mathematics, but of language (and of the usual practice and
presupposition in "word problems" ). I am not even sure the problem would be
exacly the same in French (not to mention Chinese :-)
>
>
> ******************
>
> David C. Ullrich
Denis Feldmann
> On Sun, 13 Apr 2003 09:59:22 +0200, "Denis Feldmann"
> <denis.f...@wanadoo.fr> wrote:
>
>> Answer below, but I am mostly quoting , as I feel this is worth
>> being published twice
>>
> [...]
>>
>> Your analysis is really brillant. Thank you for having so clearly
>> (and so politely :-)) laid the real issues at stake here.
>
> Three questions about that:
>
> Was it polite for him to call me a dumb fuck?
Sorry. I may have been unclear in quoting. I was *certainly* not refering to
him, who, if you recall correcty, i have insulted myself 5 days ago
**************
Eldon Moritz wrote:
> Not if you read the statement precisely. I still think, and I still
> may be wrong that the March 31, 2003 post was the first in which I
> said fuck in sci.math.
>
> Your apology is accepted.
This is typical of your very peculiar way to read statements. You are,
indeed, a dumb fuck. Also a extremely nasty bastard. Most of all, you are
completely uninteresting. Enter killfile mode.
********************************
Which, of course, pretty much ended our relation
>
>
> Was it polite for him to state repeatedly that the only reason I
> didn't agree with him was that I can't admit it when I'm wrong?
I dont think so
>
> Can you find something impolite that I said to him _before_ he
> said those things to me?
>
Of course not. As you know, I am usually on your side :-)
David C. Ullrich
<denis.f...@wanadoo.fr> wrote:
>David C. Ullrich wrote:
>> On Sun, 13 Apr 2003 09:59:22 +0200, "Denis Feldmann"
>> <denis.f...@wanadoo.fr> wrote:
>>
>>> Answer below, but I am mostly quoting , as I feel this is worth
>>> being published twice
>>>
>> [...]
>>>
>>> Your analysis is really brillant. Thank you for having so clearly
>>> (and so politely :-)) laid the real issues at stake here.
>>
>> Three questions about that:
>>
>> Was it polite for him to call me a dumb fuck?
>
>Sorry. I may have been unclear in quoting. I was *certainly* not refering to
>him, who, if you recall correcty, i have insulted myself 5 days ago
Oh yes, I forgot about that. Must have taken what you said here wrong
somehow - never mind.
>
>**************
>Eldon Moritz wrote:
>
>> Not if you read the statement precisely. I still think, and I still
>> may be wrong that the March 31, 2003 post was the first in which I
>> said fuck in sci.math.
>>
>> Your apology is accepted.
>
>This is typical of your very peculiar way to read statements. You are,
>indeed, a dumb fuck. Also a extremely nasty bastard. Most of all, you are
>completely uninteresting. Enter killfile mode.
>********************************
>
>Which, of course, pretty much ended our relation
>>
>
>
>>
>> Was it polite for him to state repeatedly that the only reason I
>> didn't agree with him was that I can't admit it when I'm wrong?
>
>I dont think so
>
>>
>> Can you find something impolite that I said to him _before_ he
>> said those things to me?
>>
>
>Of course not. As you know, I am usually on your side :-)
>
>
>
>>> I, in particular, had not
>>> realized that the true reason of those difficulties is the ambiguity
>>> of the real-life scenarios giving rise to the initial riddle, where
>>> mathematicians like myself only see a clear-cut case of Bayesian
>>> analysis
>>>
>>>
>>
>>
>> ******************
>>
>> David C. Ullrich
>
******************
David C. Ullrich
David C. Ullrich
<denis.f...@wanadoo.fr> wrote:
>David C. Ullrich wrote:
>> On Sun, 13 Apr 2003 09:59:22 +0200, "Denis Feldmann"
>> <denis.f...@wanadoo.fr> wrote:
>>
>>> Answer below, but I am mostly quoting , as I feel this is worth
>>> being published twice
>>>
>>> Kevin Buhr wrote:
>>>> David C. Ullrich <ull...@math.okstate.edu> writes:
>>>>>>
>> [...]
>>>
>>> Your analysis is really brillant. Thank you for having so clearly
>>> (and so politely :-)) laid the real issues at stake here. I, in
>>> particular, had not realized that the true reason of those
>>> difficulties is the ambiguity of the real-life scenarios giving rise
>>> to the initial riddle, where mathematicians like myself only see a
>>> clear-cut case of Bayesian analysis
>>
>> What makes you think that the question is referring to some
>> unspecified real-life scenario in the first place?
>
>Mostly because of the Monty Hall problem. There, what people were "really"
>asking was "is it smart to switch?", which cannot be decided without knowing
>the scenario.
The difference, or at least _a_ difference, is that in the Monty Hall
problem there _are_ "characters" actually appearing in the problem -
it's explicitly a problem about a game with more than one "player",
so to know what to do you need to know something about what
the other "player" is doing. As opposed to this question where
there _is_ no game being played, no experiment being run,
until it's inserted by the person reading the problem.
Or so it seems to me.
>I agree of course with you in reading the question (or the
>questions with or without "given" ; I cannot see the difference) as being a
>Bayesian question (and a trick question at that). But you must admit there
>is room for scenarios interpretation, and in fact this is the reason I never
>ask this kind of question, but questions with explicit scenarios. Anyway,
>the whole point (and here, again, I agree with you) is that all this is not
>a matter of mathematics, but of language (and of the usual practice and
>presupposition in "word problems" ). I am not even sure the problem would be
>exacly the same in French (not to mention Chinese :-)
>
>
>
>>
>>
>> ******************
>>
>> David C. Ullrich
>
******************
David C. Ullrich
Eldon Moritz
ma...@mimosa.csv.warwick.ac.uk
ull...@math.okstate.edu writes:
>On Sun, 13 Apr 2003 10:08:29 +0000 (UTC),
>ma...@mimosa.csv.warwick.ac.uk () wrote:
>
>>In article <87wuhz7...@saurus.asaurus.invalid>,
>> Kevin Buhr <bu...@telus.net> writes:
>>
>>Yes, I agree with everything you say below. I think that Eldon has been making
>>some valid points, and he does not really deserve to be treated with the
>>derision that he has received from some of the people who do not agree
>>with him.
>
>"Derision"?
>
>Have you read the entire threads? He was complaining about people
I have read most of them, but I admit that I have lost track of who
started the more extreme insults! It seems very probable that he did.
But the `derision' was not referring to you anyway - I would say that
you have been one of the more rational participants in the discussion.
For the record, I find your interpretation that the poser of the problem
was asking a purely mathematical problem on conditional probability by
far the most plausible. But I think that the `Eldon model' is a
possible alternative interpretation, although if it almost certainly not
what the problem psoer intended.
Derek Holt.
Eldon Moritz
it this way, you can never get the correct answer to the stated
question.
Stated questions do exist. This is one.
> >A good one *might be* the one I suggested,
> >but I'm by no means convinced. For example, consider the following
> >two scenarios:
> >
> >1. You are talking to a woman. She mentions she has two children,
> >but doesn't say anything else about them. Later in the conversation,
> >you ask her, "do you have a son?" She says, "yes", and seems on the
> >verge of adding either "two of them" or "he's a handful compared to
> >his sister" or who knows what when you are unfortunately interrupted.
> >
> >2. You are talking to a woman. She mentions that she has two
> >children. Then, she says, "my children are always getting into
> >trouble, especially Gregory."
> >
> >If you're a psychologist, helping the woman deal with the many
> >stresses of child-rearing, you might reasonably jot down in your
> >notebook "two children, at least one boy" in either scenario, and no
> >one would fault you for not accurately documenting the situation.
>
> Seems to me that in both of these you're inserting much
> more than is there.
>
> Especially in (2). Given that she happened to _say_ something
> that implied that at least one is a boy I don't see how we can
> reasonably deduce _anything_ about the probabilities. We'd
> need to know the answer to various poorly defined empirical
> questions, regarding who says what under what circumstances.
>
I also believe that all that is not necessary. We know where we got
our information. We got it from the problem statement. (which we were
given)
We were given a problem statement which told us of 'a' woman. Had it
not been for the statement we would not even have know that she
existed.
We have 'a' statement, about 'a' woman. She has two children. There is
evidence of 'one' boy. There is no evidence about the sex of the other
child.
> _When_ we analyze the problem using this sort of "scenario"
> it seems to me that there's simply no answer available -
> one can invent a scenario that will give whatever answer
> one wishes. Seems to me the only way one can give an
> answer to the question is to take the "at least one is a
> boy" as "given that at least one is a boy". That's assuming
> exactly what's given in the problem, not adding other
> factors.
>
I agree. If they state a scenario, then the question answers easy.
Just as, if the poser wanted to have this to be given at least one,
the poser should have said so. Invent no scenarios, don't speculate on
what the poser intended, only what the poser said.
We have a problem statement about a woman. There was no scenario in
the statement.
But there is a woman.
We know from the statement that she doesn't have GG.
She has BB, and the statement was made, OR,
She has BG, and the statement was made, OR,
She has GB, and the statement was made.
Here, we agree.
> If the problem _were_ describing the result of some process
> whereby we obtained the information, then yes indeed we'd
> need to know much more about _how_ we obtained this
> information - that's so true that if the problem _were_ about
> the result of an experiment then the answer is clearly just
> that there's not enough information given to answer the
> question. Hence the problem must not be about the
> result of some experiment.
>
I agree, but suppose that our woman has BB. Bill is betting that she
has two, Joe is betting that she has one of each. Bill will win. How
do you explain to Joe that he has to pay two to one How do you assure
him that, had the woman had GG, you wouldn't have done the same thing
in reverse, and he'd have to pay two to one there also? All Bill and
Joe hear is the statement.
Eldon
Eldon Moritz
> In article <foni9vg3100914oii...@4ax.com>,
> ull...@math.okstate.edu writes:
> >On Sun, 13 Apr 2003 10:08:29 +0000 (UTC),
> >ma...@mimosa.csv.warwick.ac.uk () wrote:
> >
> >>In article <87wuhz7...@saurus.asaurus.invalid>,
> >> Kevin Buhr <bu...@telus.net> writes:
> >>
> >>Yes, I agree with everything you say below. I think that Eldon has been making
> >>some valid points, and he does not really deserve to be treated with the
> >>derision that he has received from some of the people who do not agree
> >>with him.
> >
> >"Derision"?
> >
> >Have you read the entire threads? He was complaining about people
>
> I have read most of them, but I admit that I have lost track of who
> started the more extreme insults! It seems very probable that he did.
> But the `derision' was not referring to you anyway - I would say that
> you have been one of the more rational participants in the discussion.
> For the record, I find your interpretation that the poser of the problem
> was asking a purely mathematical problem on conditional probability by
> far the most plausible. But I think that the `Eldon model' is a
> possible alternative interpretation, although if it almost certainly not
> what the problem psoer intended.
>
> Derek Holt.
>
he started, really used it, but that's okay. I had read in one of his
earlier posts where he used the 'f' word so I knew that he was
familiar with the term. I don't think I've actually called anyone else
stupid. (I probably thought it and it shined through) The important
thing is to get this damn question answered correctly.
This question was in a family magazine, it wasn't an abstract
mathematical question. It's a simple little stated question. I don't
know whetehr Ullrich's interpretation is plausible, or not, I know
it's wrong.
Probability for two, given at least one is a mathematical term. It
defines what it means to say "given at least one", at least in a
mathematical forum. Outside of a math forum, putting the word 'given'
in a question might need some explaining.
"Given at least one" means that we take all the women for which the
statement would be true. It establishes a three statements to four
women ratio. It says that we demur on the other two of a kind. For a
statement to bettors, the statement must be made prior to the final
selection of the woman.
The definition does not define the statement "at least one is." The
statement "at least one is" can only be made after inspection.
That's the KEY. The KEY. I've talked about the key, and no one seems
to notice. The Key to getting thirds, is to make the boy/girl decision
prior to selection.
The Key to getting fourths, is to select, prior to making the boy/girl
decision.
Select with no prejudice, or flip two coins without prejudice, make
the inspection and it's one of four. It's a fourth of something.
Flip two coins, then look. You're stuck with 1 of four. You can't make
a statement about one of them that will alter the two bettor's odds.
(okay, maybe you could say, "this boy has a sister" or something. I
mean a mathematically equivalent statement. The statement whereas the
woman was selected, and the statement was made, has correct answer
1/2.
See "at least one is" and jump directly to given at least one, and you
have erred. You can never get the correct answer to my question.
Mathematics does not have that kind of gaps. It doesn't have little
simple questions with no answers. You must be able to answer my
question.
When you can correctly answer my question, then, you'll see that the
question for given at least one has to be different. We have a
statement with one statement to one woman ratio. To get correct
answer, you must change this ratio.
I'm not worried about my treatment. I was a professional baseball
umpire, I enjoy being around people who hate me. The important thing
is to understand my logic.
Eldon
David C. Ullrich
wrote:
>David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<vini9vk1164atne60...@4ax.com>...
>> On Sun, 13 Apr 2003 09:59:22 +0200, "Denis Feldmann"
>> <denis.f...@wanadoo.fr> wrote:
>>
>> >Answer below, but I am mostly quoting , as I feel this is worth being
>> >published twice
>> >
>> >Kevin Buhr wrote:
>> >> David C. Ullrich <ull...@math.okstate.edu> writes:
>> >>>>
>> [...]
>> >
>> >Your analysis is really brillant. Thank you for having so clearly (and so
>> >politely :-)) laid the real issues at stake here. I, in particular, had not
>> >realized that the true reason of those difficulties is the ambiguity of the
>> >real-life scenarios giving rise to the initial riddle, where mathematicians
>> >like myself only see a clear-cut case of Bayesian analysis
>>
>> What makes you think that the question is referring to some
>> unspecified real-life scenario in the first place?
>>
>If mathematics can't answer real life questions, what is it good for?
You really never tire of exhibiting your stupidity.
Saying that a certain question is not about a real-life situation
does not say that mathematics can't answer real-life questions.
I mean this is really really stupid. Nice work.
>Eldon
>
>
>>
>> ******************
>>
>> David C. Ullrich
******************
David C. Ullrich
David C. Ullrich
wrote:
>David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<mjmi9vggfi496jib6...@4ax.com>...
>> On Sat, 12 Apr 2003 22:55:02 GMT, Kevin Buhr <bu...@telus.net> wrote:
[...]
>
>Stated questions do exist. This is one.
What a blessed idiot:
"Stated questions do exist."
"Oh, stated questions do exist? I didn't realize that. But
come to think of it you're right, stated questions do exist.
So the answer to that question must be 1/2. I never
looked at it that way, thanks."
Who the heck has denied that "stated questions do exist"?
How does that say anything about what the _answer_ to
a certain stated question is?
You're better off restricting your replies to clever one-liners
like "dumb fuck". Because then it doesn't look like your
_trying_ to make sense - here it looks like you're trying to
prove something by stating that stated questions exist,
and it comes off utterly hilarious.
******************
David C. Ullrich
Eldon Moritz
> ull...@math.okstate.edu writes:
> >On Sun, 13 Apr 2003 10:08:29 +0000 (UTC),
> >ma...@mimosa.csv.warwick.ac.uk () wrote:
> >
> >>In article <87wuhz7...@saurus.asaurus.invalid>,
> >> Kevin Buhr <bu...@telus.net> writes:
> >>
> >>Yes, I agree with everything you say below. I think that Eldon has been making
> >>some valid points, and he does not really deserve to be treated with the
> >>derision that he has received from some of the people who do not agree
> >>with him.
> >
> >"Derision"?
> >
> >Have you read the entire threads? He was complaining about people
>
> I have read most of them, but I admit that I have lost track of who
> started the more extreme insults! It seems very probable that he did.
> But the `derision' was not referring to you anyway - I would say that
> you have been one of the more rational participants in the discussion.
> For the record, I find your interpretation that the poser of the problem
> was asking a purely mathematical problem on conditional probability by
> far the most plausible. But I think that the `Eldon model' is a
> possible alternative interpretation, although if it almost certainly not
> what the problem psoer intended.
>
> Derek Holt.
>
You have said that you think it's absurd that I think my
interpretation is the only correct interpretation. I've said a lot,
and probably have said that at one time or another. My main insistance
has been that I have a working model which works, and that they don't.
I've said, "Show me a working model that works, and I'll capitulate."
So far no takers.
We started in the counter-intuitive thread. I advanced problem A, "Two
coins were flipped, given that at least one is a head, what are the
chances for two heads?" and problem B. "Two coins were flipped, and at
least one is a tail."
At Ullrich's insistance, we went to A' and B', whereas B' is "two
coins were flipped, and at least one is a head."
Ullrich made some snide comments about me being tricky and that
changing from heads to tails certainly didn't change the question.
Notice that me made agreement here, Going From Heads To Tails
certainly does not change the question, or the probabilities. (call
this agreement a)
I started this Abstract Mathematical Question thread because Ullrich
had introduced the term. Notice that in the first post, I asked, "What
is the definition of an Abstract mathematical question and how do we
know when we have one?" and that so far, everyone seems to have
answered that.
Also, in this first post I re introduced math models 1 & 2; also
Question Q.
You said I have made some good points. Have I made a point, whereas
you can say, "That point is wrong"? Have I made a false statement,
other than the assertion that my 'interpretion' is the correct
'interpretation'?
If my assertion is absurd, my argument must be wrong somewhere. Where?
I have said that the P(A|B)=1/3 formula defines "given at least one
is" but it doesn't define "at least one is". I've said that several
times, no one seems to have taken notice, yet; Professor Ullrich does
exactly that, and you say that is "the most plausible" interpretation.
I have asked, "If the poser thinks one question, then writes another,
which question should the answerer answer?" No one seems to have
noticed that either.
THE ANSWERER ONLY SEES THE WRITTEN QUESTION.
I have introduced Bill and Joe. Two bettors. No one seems to have
noticed them. It's a 'given' that Bill bets for two of the same. Joe
bets for one of each. They only hear, or see, the written question.
All they have to do to make the proper bet, is answer the question
correctly. All we have to do to answer the question correctly, is to
correctly figure their odds.
Two coins were flipped. They landed TT. Bill will win. What odds
should Joe pay? or; Two coins were flipped. They landed HH. Bill will
win. What odds should Joe pay?
Or, Joe wins with HT, or TH. Whatever money Bill put up, Joe can take.
Joe wins with one of each, Bill wins with two of a kind, that's a
fifty fifty.
Math model 1 defines what it means to say "given at least one."
Math model 2 defines what it means to say "at least one is."
They're different.
I asked several times, "If a student asks, what does it mean to say,
'given at least one head?'" What should the answer be. No one seems to
want to answer that.
The answer, is the hypothis for model 1.
The heads decision was made prior to the flip, then flipped with
extreme prejudice toward heads.
In model 2, the flip was made without prejudice, then the "at least
one is a heads" statement was made with "at least one is a tail"
having been equally likely. We agreed earlier that changing from
"tails" to heads here is no problem. That's why we went from B to B'.
We could have gone from B' to B, just as easily.
So, we have models 1 & 2. Then we have a statement "at least one is"
and we don't know what to do? We go to model 1, and that's the most
plausible? We go to 1 because that's the definition of "given at least
one" and we're trying to decide, our primary question for this long
debate was, does the addition of "given" into the question change the
question? The quality of mathematics hasn't deteriorated that far.
A'was "given at least one", B' was "at least one". Our question was,
"are they different?" Think about that. Ullrich jumps to the answer
for A' because he thinks that's the answer the poser thought?
Look at it this way. If A' and B' are the same, then they are both "at
least one is". They should both be answered as stated questions. A'
was correctly answered 1/3 because the word 'given' made it an
'abstract mathematical question' (whatever that is), then B' was
answered 1/3 because it's like A'. And that's the most plausible?
....Nope.....
B' has correct answer iff 'heads' were selected prior to the flip.
There are some ambiguities in arguments. We can discuss "prior to the
flip" and such, and we can set the definitions. There is no ambiguity
in B'.
Bayesians are not stuck to their own little room, destined to
incorrect answers to certain questions, unless they choose to be. The
Bayes' formulas are distinct, and correct, the math works. The
Bayseian who will take the effort to put the correct numbers in will
get correct answers. Otherwise, garbage in, garbage out.
Model 1 defines "given at least one". Alter the numbers in model 1 and
you can change the definition. It doesn't make sense? It doesn't have
to. In math we can define things however we want them. Alter it to
hell and back, it doesn't change the answer to B'.
Ullrich has denied my argument without apparent inspection. That isn't
plausible. You have seen some of my good 'points'. Show me a bad one.
Show me where I've erred. If I've erred, I'll capitulate. Just show me
where it is.
Two coins were flipped and at least one is a head. When the flipper
did exactly what the statement said, then the correct answer is 1/2.
On BB, Joe has no reason to pay two to one. There is nothing in the
statement that tells him he would get a demur on TT.
Eldon
ma...@mimosa.csv.warwick.ac.uk
elmo...@yahoo.com (Eldon Moritz) writes:
>ma...@mimosa.csv.warwick.ac.uk () wrote in message news:<b7cmob$k85$1...@wisteria.csv.warwick.ac.uk>...
>> In article <foni9vg3100914oii...@4ax.com>,
>> ull...@math.okstate.edu writes:
>> >On Sun, 13 Apr 2003 10:08:29 +0000 (UTC),
>> >ma...@mimosa.csv.warwick.ac.uk () wrote:
>> >
OK, I will answer this one, but please forgive me if I decide to make
this my last contribution to this thread! I think that everything there
is to be said about it has been said many times already.
>
>You have said that you think it's absurd that I think my
>interpretation is the only correct interpretation. I've said a lot,
>and probably have said that at one time or another. My main insistance
>has been that I have a working model which works, and that they don't.
>I've said, "Show me a working model that works, and I'll capitulate."
>So far no takers.
Two other working models that work are:
1. The question B is to be interpreted as meaning the same as the
mathematical question A. The meaning is equivalent to the same question,
but using the word 'given' and so the answer of 1/3 is the same.
2. We use your observer model but biased towards heads. So, if the
observer sees two tails, he says "at least one coin is tails" or
maybe he says nothing at all, but otherwise he says "at least one coin
is heads". This also gives the answer 1/3.
I have said that I find 1, the most plausible and convincing interpretation,
so my vote goes for that one. I find 2. less good than your model, but that
still does not say that interpretation 2. is wrong. This is really where
we disagree. There is no single correct answer. As far as I can see, there
is no disagreement between you and me and Ullrich and Buhr and any of the
others about the analysis of the possible mathematical modules. The
disagreement is purely linguistic, and is about interpretations of
statements that contain incomplete information. If a statement does not
contain sufficient information, then there is no single correct
interpretation.
A problem in probability might just be a purely mathematical problem
disguised as a hypothetical real-life incident. I think that it is highly
likely that this is the case here.
Or it might be about a real experiment that has taken place or is about
to take place. If it is about an experiment, then we can only answer the
problem if we are given complete details about how the experiment is
carried out, and we must be told what will happen and what will be
reported under all possible outcomes of the experiment. If the problem
here is really about such an experiment (which does not seem likely) then we
have incomplete information, and so the problem is ambiguous and there is
no single correct answer.
Either way, 1/2 cannot be the unique correct answer.
>We started in the counter-intuitive thread. I advanced problem A, "Two
>coins were flipped, given that at least one is a head, what are the
>chances for two heads?" and problem B. "Two coins were flipped, and at
>least one is a tail."
>
>At Ullrich's insistance, we went to A' and B', whereas B' is "two
>coins were flipped, and at least one is a head."
yes ...
>I started this Abstract Mathematical Question thread because Ullrich
>had introduced the term. Notice that in the first post, I asked, "What
>is the definition of an Abstract mathematical question and how do we
>know when we have one?" and that so far, everyone seems to have
>answered that.
>Also, in this first post I re introduced math models 1 & 2; also
>Question Q.
>
>You said I have made some good points. Have I made a point, whereas
>you can say, "That point is wrong"? Have I made a false statement,
>other than the assertion that my 'interpretion' is the correct
>'interpretation'?
Your false statement is that your interpretation or model is the unique
correct one. There is nothing wrong with the model itself.
>If my assertion is absurd, my argument must be wrong somewhere. Where?
I have not seen an argument that your interpretation is the *only* possible
correct one.
>I have said that the P(A|B)=1/3 formula defines "given at least one
>is" but it doesn't define "at least one is". I've said that several
>times, no one seems to have taken notice, yet; Professor Ullrich does
>exactly that, and you say that is "the most plausible" interpretation.
There are people who believe that whoever asked the question intended the
question to mean the same thing as "given at least one is ..."
Yes, on balance, I find that the most plausible. I think it extremely
unlikely that any real coins were actually tossed. So the problem poser
was not reporting on any events that had actually taken place, but was
hypothesizing the whole situation. Since the whole scenario was only
happening within the mind of the problem poser, the most sensible
interpretation has to be the one intended by the poser, which I believe
to be the "given at least ..." question.
>I have asked, "If the poser thinks one question, then writes another,
>which question should the answerer answer?" No one seems to have
>noticed that either.
Yes I noticed that nobody had responded to that one! It is an interesting
and debatable question. Again, on balance, I think it might be better
and more constructive to answer the intended question, but also to point out
any ambiguities. If you were taking an exam, then you would be best
advised to answer the intended question I think!
>THE ANSWERER ONLY SEES THE WRITTEN QUESTION.
Yes, but in some situations, like this one, they can also make a very
informed guess about what the intended question might have been!
>I have introduced Bill and Joe. Two bettors. No one seems to have
>noticed them. It's a 'given' that Bill bets for two of the same. Joe
>bets for one of each. They only hear, or see, the written question.
>All they have to do to make the proper bet, is answer the question
>correctly. All we have to do to answer the question correctly, is to
>correctly figure their odds.
>
>Two coins were flipped. They landed TT. Bill will win. What odds
>should Joe pay? or; Two coins were flipped. They landed HH. Bill will
>win. What odds should Joe pay?
>
>Or, Joe wins with HT, or TH. Whatever money Bill put up, Joe can take.
>
>Joe wins with one of each, Bill wins with two of a kind, that's a
>fifty fifty.
Yes.
>Math model 1 defines what it means to say "given at least one."
>Math model 2 defines what it means to say "at least one is."
This is your definition. Others do not agree with these definitions.
In the end, it is just a matter of opinion.
>They're different.
>
>I asked several times, "If a student asks, what does it mean to say,
>'given at least one head?'" What should the answer be. No one seems to
>want to answer that.
As far as I know, nobody has argued about what "given at least one head"
means.
>The answer, is the hypothis for model 1.
>
>The heads decision was made prior to the flip, then flipped with
>extreme prejudice toward heads.
>
>In model 2, the flip was made without prejudice, then the "at least
>one is a heads" statement was made with "at least one is a tail"
>having been equally likely. We agreed earlier that changing from
>"tails" to heads here is no problem. That's why we went from B to B'.
>We could have gone from B' to B, just as easily.
>
>So, we have models 1 & 2. Then we have a statement "at least one is"
>and we don't know what to do? We go to model 1, and that's the most
>plausible? We go to 1 because that's the definition of "given at least
>one" and we're trying to decide, our primary question for this long
>debate was, does the addition of "given" into the question change the
>question? The quality of mathematics hasn't deteriorated that far.
It has nothing whatsoever to do with the quality of mathematics - it is
a purely linguistic matter. Many people believe that the addition of
"given" does not change the question.
You seem to be continually trying to say that there is a mathematical issue
involved in the dispute. There in none! The dispute is purely linguistic,
or maybe philosophical.
>A'was "given at least one", B' was "at least one". Our question was,
>"are they different?" Think about that. Ullrich jumps to the answer
>for A' because he thinks that's the answer the poser thought?
That seems very reasonable to me!
>Look at it this way. If A' and B' are the same, then they are both "at
>least one is". They should both be answered as stated questions. A'
>was correctly answered 1/3 because the word 'given' made it an
>'abstract mathematical question' (whatever that is), then B' was
>answered 1/3 because it's like A'. And that's the most plausible?
>....Nope.....
I find that the most plausible, yes.
>B' has correct answer iff 'heads' were selected prior to the flip.
>There are some ambiguities in arguments. We can discuss "prior to the
>flip" and such, and we can set the definitions. There is no ambiguity
>in B'.
As I said earlier, maybe 'heads' was selected prior to the flip. That
would indicate a bias, but there is already a bias in that the problem
mention 'heads' and does not mention 'tails', so you cannot rule out
the possibility of a bias. Even if you find that argument weak, you cannot
assert that it is wrong, you can only say that it is weak or implausible.
>Bayesians are not stuck to their own little room, destined to
>incorrect answers to certain questions, unless they choose to be. The
>Bayes' formulas are distinct, and correct, the math works. The
>Bayseian who will take the effort to put the correct numbers in will
>get correct answers. Otherwise, garbage in, garbage out.
>
>Model 1 defines "given at least one". Alter the numbers in model 1 and
>you can change the definition. It doesn't make sense? It doesn't have
>to. In math we can define things however we want them. Alter it to
>hell and back, it doesn't change the answer to B'.
Exactly. Some mathematicians define the set N of natural numbers to
contain 0, and some define it not to contain 0. Are you saying that one
of these definitions is right and the other wrong? You have to accept that
not everybody is using the same definitions as you are. So once again, the
dispute is linguistic.
Derek Holt.
ma...@mimosa.csv.warwick.ac.uk
ma...@mimosa.csv.warwick.ac.uk () writes:
>In article <349f5619.03041...@posting.google.com>,
> elmo...@yahoo.com (Eldon Moritz) writes:
>>ma...@mimosa.csv.warwick.ac.uk () wrote in message news:<b7cmob$k85$1...@wisteria.csv.warwick.ac.uk>...
>>> In article <foni9vg3100914oii...@4ax.com>,
>>> ull...@math.okstate.edu writes:
>>> >On Sun, 13 Apr 2003 10:08:29 +0000 (UTC),
>>> >ma...@mimosa.csv.warwick.ac.uk () wrote:
>>> >
>...
>
PS to last post
Eldon:
I meant to say also that there is another serious weakness in your unbiased
model for the "at least one is a head" problem, in that it does not
generalize in any obvious way to more complicated problems. How about:
There are 6 balls, which are red, green or yellow with equal probability,
and at least one ball is red. What is the probability that all non-red
balls are green?
What is your unbiased model for this question? You may come up with an
answer, but to be consistent, you will have to demonstrate that your
model is the only possible correct one, and that any other model is just
wrong. I can think of at least two possibilities which would give different
answers.
I am not attacking your model in particular, I am only disputing your claim
that it is the only possible correct model.
Derek Holt.
Eldon Moritz
> On 13 Apr 2003 18:36:34 -0700, elmo...@yahoo.com (Eldon Moritz)
> wrote:
>
> >David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<mjmi9vggfi496jib6...@4ax.com>...
> >> On Sat, 12 Apr 2003 22:55:02 GMT, Kevin Buhr <bu...@telus.net> wrote:
> [...]
> >
> >Stated questions do exist. This is one.
>
> What a blessed idiot:
>
> "Stated questions do exist."
>
> "Oh, stated questions do exist? I didn't realize that. But
> come to think of it you're right, stated questions do exist.
> So the answer to that question must be 1/2. I never
> looked at it that way, thanks."
>
questions do exist. This should mean that you agree with the parts you
passed over? In that case, my argument stands and we agree one the
answer 1/2.
> Who the heck has denied that "stated questions do exist"?
> How does that say anything about what the _answer_ to
> a certain stated question is?
>
There is a math model which defines the probability for two, given at
least one. By your own admission, when you see "at least one is" you
go directly to this model, no matter what Eldon says. You deny Eldon's
question to go to a mathematical definition.
Yet, you refuse to even say what the definition is. I've asked you
repeatedly, "What does it mean to say given at least one?" If you've
answered, I haven't seen it yet.
There is an answer.
Eldon
Eldon Moritz
> In article <349f5619.03041...@posting.google.com>,
> elmo...@yahoo.com (Eldon Moritz) writes:
> >ma...@mimosa.csv.warwick.ac.uk () wrote in message news:<b7cmob$k85$1...@wisteria.csv.warwick.ac.uk>...
> >> In article <foni9vg3100914oii...@4ax.com>,
> >> ull...@math.okstate.edu writes:
> >> >On Sun, 13 Apr 2003 10:08:29 +0000 (UTC),
> >> >ma...@mimosa.csv.warwick.ac.uk () wrote:
> >> >
> ...
>
> OK, I will answer this one, but please forgive me if I decide to make
> this my last contribution to this thread! I think that everything there
> is to be said about it has been said many times already.
>
questions which haven't been answered. I've left several arguments out
there unchallenged. The length of the thread may be a problem. It
ain't my problem.
> >
> >You have said that you think it's absurd that I think my
> >interpretation is the only correct interpretation. I've said a lot,
> >and probably have said that at one time or another. My main insistance
> >has been that I have a working model which works, and that they don't.
> >I've said, "Show me a working model that works, and I'll capitulate."
> >So far no takers.
>
> Two other working models that work are:
>
A system of flipping the coins, or selecting the families, which fits
into the confines of our problem statement. Computerized, or
otherwise. Imaginary, or hypothetical. I do a lot of imaginary flips,
hypothetically the ratios come out exactly. The rules of the game
don't change. Heads and tails must be equally likely, and the problem
statement must be true.
One model:
Our computer was programmed so that all the flipper, or poser sees is
a set of distinguishable lights.
There was a red light and a green light. At least one is green, or at
least one is red are both true statements. (this would suffice as a
pretty good definition of "at least one". To say given at least one
green here, wouldn't mean much with our current definition of given at
least one)
The poser chose a light, the computer wrote, "Two coins were flipped
and at least one is a head. What are the chances for two heads?" Or,
"A woman has two children and at least one is a boy. What are the
chances for two boys?"
That's a working model and the correct answer to either question is
1/2. The one thirders need more information.
> 1. The question B is to be interpreted as meaning the same as the
> mathematical question A. The meaning is equivalent to the same question,
> but using the word 'given' and so the answer of 1/3 is the same.
>
"The question B is to be interpreted as meaning the same as..." Did
you mean this to be part of your question? If so, it adds words.
Our original question, for the recent argument was, "Are A and B the
same? You said, interpret B as meaning the same, then they are the
same? That's our mystery. Of course, if we interpret them to be the
same, they are the same.
Shame, shame, Doc, that's not mathematically adequate.
> 2. We use your observer model but biased towards heads. So, if the
> observer sees two tails, he says "at least one coin is tails" or
> maybe he says nothing at all, but otherwise he says "at least one coin
> is heads". This also gives the answer 1/3.
>
That's my argument. To get a correct answer 1/3, we must have a bias,
or prejudice, or prior choice. That's the KEY to the question. Do we
have a bias or not. (Dr. Gray told me that bias is also a special
mathematical word, so for this case I should use one of the others. He
suggested extreme prejudice)
We can make a model which gets correct answer 1/3. We can't explain
the workings of the model without adding words to the question. In my
model, all we know is that there were two, we don't know which was
which. The poser chose one and the question spit out. To ADD a BIAS
has to add words, information. To get correct answer 1/3, information,
words, have to be added. The only communication between poser, and
answerer is the problem statement. The bias has to be communicated
through our problem statement. "Two coins were flipped, until at least
one is a heads." The "until" communicates this bias. It adds a little
information. Information necessary to the correct 1/3.
> I have said that I find 1, the most plausible and convincing interpretation,
> so my vote goes for that one. I find 2. less good than your model, but that
> still does not say that interpretation 2. is wrong. This is really where
> we disagree. There is no single correct answer. As far as I can see, there
> is no disagreement between you and me and Ullrich and Buhr and any of the
> others about the analysis of the possible mathematical modules. The
> disagreement is purely linguistic, and is about interpretations of
> statements that contain incomplete information. If a statement does not
> contain sufficient information, then there is no single correct
> interpretation.
Ullrich says that there is one answer, it's a third. He says that my
argument is wrong, and has no merit. He says that I'm completely wrong
because ya'll have already decided, with a concencuous, that "at least
one is" means "given, at least one." He uses that excuse to
completely ignore my argument. Buhr seemed to be considering my
argument, until Ullrich intercepted him. Don't think we've heard from
him lately. Buhr and I may have agreement.
You and I have agreement, up to a point. Then you stop considering my
argument.
>
We don't disagree on the workings of the models. We disagree on what
it means to say "at least one is". We disagree on what it means to say
"given at least one", and maybe not, as no one has actually stated
this interpretation. The meaning of given at least one is that, with
given at least one, there is a bias. Given at least one introduces a
bias.
We all seem to agree that "given at least one" introduces a bias.
Ya'll don't seem to agree that when WE, the answerers of the question
go from "at least one" to "given at least one", then WE have
INTRODUCED the bias. When we introduce the bias, then we change the
question.
Linguistics are taught in the English Lit dept. Mathematicians should
take a course or two.
> A problem in probability might just be a purely mathematical problem
> disguised as a hypothetical real-life incident. I think that it is highly
> likely that this is the case here.
>
We are talking about the stated question, "Two coins were flipped, and
at least one is a head. What are the chances for two heads?" or "A
woman has two children, and at least one is a boy. What are her
chances for two boys?" Call this, or these, Our Question.
Both have appeared through the years, inside and outside of the halls
of mathematics. If there is a difference between being inside, or
outside, then the onus is on the mathematician to explain the
difference when going outside. The mathematician should understand the
difference, it should be explained in math classes.
We are talking about a stated question. They do exist. Our Question is
a stated question. Basically, in a stated question, ALL WE HAVE is THE
STATEMENT.
> Or it might be about a real experiment that has taken place or is about
> to take place. If it is about an experiment, then we can only answer the
> problem if we are given complete details about how the experiment is
> carried out, and we must be told what will happen and what will be
> reported under all possible outcomes of the experiment. If the problem
> here is really about such an experiment (which does not seem likely) then we
> have incomplete information, and so the problem is ambiguous and there is
> no single correct answer.
>
> Either way, 1/2 cannot be the unique correct answer.
>
Ullrich hasn't agreed that we have an ambiguous statement. He says
it's 1/3.
I said, my working model works. So far, you haven't shown me a working
model which does.
> >We started in the counter-intuitive thread. I advanced problem A, "Two
> >coins were flipped, given that at least one is a head, what are the
> >chances for two heads?" and problem B. "Two coins were flipped, and at
> >least one is a tail."
> >
> >At Ullrich's insistance, we went to A' and B', whereas B' is "two
> >coins were flipped, and at least one is a head."
>
> yes ...
>
> >I started this Abstract Mathematical Question thread because Ullrich
> >had introduced the term. Notice that in the first post, I asked, "What
> >is the definition of an Abstract mathematical question and how do we
> >know when we have one?" and that so far, everyone seems to have
> >answered that.
> >Also, in this first post I re introduced math models 1 & 2; also
> >Question Q.
> >
> >You said I have made some good points. Have I made a point, whereas
> >you can say, "That point is wrong"? Have I made a false statement,
> >other than the assertion that my 'interpretion' is the correct
> >'interpretation'?
>
> Your false statement is that your interpretation or model is the unique
> correct one. There is nothing wrong with the model itself.
>
I have a working model which works. I have an argument which shows
that the correct answer is 1/2. My argument pretty well shows that
other arguments are wrong.
My 'interpretation that 1/2 is unique' is not a my false statement. It
is my conclusion. Don't just say that my conclusion is false. Knock
out one of my points. Disprove 'a' point in my argument.
> >If my assertion is absurd, my argument must be wrong somewhere. Where?
>
> I have not seen an argument that your interpretation is the *only* possible
> correct one.
>
Follow my argument through.
> >I have said that the P(A|B)=1/3 formula defines "given at least one
> >is" but it doesn't define "at least one is". I've said that several
> >times, no one seems to have taken notice, yet; Professor Ullrich does
> >exactly that, and you say that is "the most plausible" interpretation.
>
> There are people who believe that whoever asked the question intended the
> question to mean the same thing as "given at least one is ..."
> Yes, on balance, I find that the most plausible. I think it extremely
> unlikely that any real coins were actually tossed. So the problem poser
> was not reporting on any events that had actually taken place, but was
> hypothesizing the whole situation. Since the whole scenario was only
> happening within the mind of the problem poser, the most sensible
> interpretation has to be the one intended by the poser, which I believe
> to be the "given at least ..." question.
>
We agree that there are people who believe that the poser wrote one
question and intended another.
I answer the question which the poser wrote. I don't get into
'intended'. That's where the ambiguity comes in. There are a lot of
ambiguities in figuring out the intended.
We agree that you believe whatever it is you say you believe. But, in
a written statement, all we have is the statement.
As WRITTEN, our question has correct answer 1/2. As intended? It has
many answers, to this we have agreement.
**************
Then we have YOUR statement: "I think it extremely unlikely that any
real coins were actually tossed".
************
This is the crux of the disagreement. This is an extremely important
point. This is probably where you stray from my argument, where you
refuse to understand my argument. (YOU, here meaning the complete one
thirder community)
This may be where this argument should be argued. If this thread's too
long, maybe we need a new thread. THIS IS SIGNIFICANT.
Our statement states, "TWO COINS WERE TOSSED..." "'A' woman HAS two
children..." It says that right there in Our Question.
We are talking about 'the' results of a two coin toss, or we are
talking about the family of 'A' woman.
I rest my case.
Eldon Moritz
David C. Ullrich
wrote:
>David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<of0l9v0mkkm4phhd1...@4ax.com>...
>> On 13 Apr 2003 18:36:34 -0700, elmo...@yahoo.com (Eldon Moritz)
>> wrote:
>>
>> >David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<mjmi9vggfi496jib6...@4ax.com>...
>> >> On Sat, 12 Apr 2003 22:55:02 GMT, Kevin Buhr <bu...@telus.net> wrote:
>> [...]
>> >
>> >Stated questions do exist. This is one.
>>
>> What a blessed idiot:
>>
>> "Stated questions do exist."
>>
>> "Oh, stated questions do exist? I didn't realize that. But
>> come to think of it you're right, stated questions do exist.
>> So the answer to that question must be 1/2. I never
>> looked at it that way, thanks."
>>
>You bypassed quite a bit of my argument, to agree with me that stated
>questions do exist. This should mean that you agree with the parts you
>passed over?
Dumb fuck.
>In that case, my argument stands and we agree one the
>answer 1/2.
>
>
>> Who the heck has denied that "stated questions do exist"?
>> How does that say anything about what the _answer_ to
>> a certain stated question is?
>>
>There is a math model which defines the probability for two, given at
>least one. By your own admission, when you see "at least one is" you
>go directly to this model, no matter what Eldon says. You deny Eldon's
>question to go to a mathematical definition.
You know it really does look very strange when you refer to yourself
in the third person this way.
Yes, it's true that when I read the question I go directly to an
analysis of the question based on the meaning of what was
written, "bypassing" your arguments that it means something
else, no matter what you say. Just as if the question was
"You have two apples. Someone gives you two more. How
many apples do you have?" I would go directly to the model
2 + 2 = 4, no matter what you said about how we need to
consider the question of how many other apples someone
else might have donated that were not mentioned in the
problem, no matter what you said about how we need to
consider how we determined that two more apples
were donated, no matter what you say about a lot
of totally irrelevant considerations.
You should _publish_ your work on this question!
I mean you're one of the foremost experts on the
problem, perhaps _the_ expert, according to that
"paper" you wrote with Gray. Usually the foremost
expert on something is willing to publish his
results.
>Yet, you refuse to even say what the definition is. I've asked you
>repeatedly, "What does it mean to say given at least one?" If you've
>answered, I haven't seen it yet.
>
>There is an answer.
>
>Eldon
>
>> You're better off restricting your replies to clever one-liners
>> like "dumb fuck". Because then it doesn't look like your
>> _trying_ to make sense - here it looks like you're trying to
>> prove something by stating that stated questions exist,
>> and it comes off utterly hilarious.
>>
>>
>>
>>
>> ******************
>>
>> David C. Ullrich
******************
David C. Ullrich
Eldon Moritz
call it my question, it doesn't matter what I call it, or how strange
it looks. You can't answer it, then you call me Dumb Fuck. That's all
you're left with, because the truth is on my side.
> Yes, it's true that when I read the question I go directly to an
> analysis of the question based on the meaning of what was
> written, "bypassing" your arguments that it means something
> else, no matter what you say. Just as if the question was
> "You have two apples. Someone gives you two more. How
> many apples do you have?" I would go directly to the model
> 2 + 2 = 4, no matter what you said about how we need to
> consider the question of how many other apples someone
> else might have donated that were not mentioned in the
> problem, no matter what you said about how we need to
> consider how we determined that two more apples
> were donated, no matter what you say about a lot
> of totally irrelevant considerations.
>
We have two questions differing only by the word given. You say they
are the same.
B' = x.
A' = given + x.
You say A' and B' are the same.
given + x = x
given = 0
We have math model 1
It defines given.
There is a bias.
model 2 has no bias.
B' has no bias.
There, you use "given" to establish a bias, even though it means
nothing.
Then when you get question B', which shows no bias, but, you say it's
the same as A' therefore it has the same bias.
Given can't be zero, and introduce a bias.
Your logic sucks.
> You should _publish_ your work on this question!
> I mean you're one of the foremost experts on the
> problem, perhaps _the_ expert, according to that
> "paper" you wrote with Gray. Usually the foremost
> expert on something is willing to publish his
> results.
>
On this question, I'll go point to point with you, and you can't win.
The truth is on my side.
> >Yet, you refuse to even say what the definition is. I've asked you
> >repeatedly, "What does it mean to say given at least one?" If you've
> >answered, I haven't seen it yet.
> >
> >There is an answer.
> >
> >Eldon
> >
> >> You're better off restricting your replies to clever one-liners
> >> like "dumb fuck". Because then it doesn't look like your
> >> _trying_ to make sense - here it looks like you're trying to
> >> prove something by stating that stated questions exist,
> >> and it comes off utterly hilarious.
> >>
You're left with the "dumb fucks" because the truth is on my side.
Do you fear the truth?
Eldon
David C. Ullrich
wrote:
>David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<d5er9v8dirvq8oirn...@4ax.com>...
>> On 16 Apr 2003 06:36:39 -0700, elmo...@yahoo.com (Eldon Moritz)
>> wrote:
>>
[...]
>> >>
>> >There is a math model which defines the probability for two, given at
>> >least one. By your own admission, when you see "at least one is" you
>> >go directly to this model, no matter what Eldon says. You deny Eldon's
>> >question to go to a mathematical definition.
>>
>> You know it really does look very strange when you refer to yourself
>> in the third person this way.
>>
>There is a question which you can't answer. Call it Eldon's question,
>call it my question, it doesn't matter what I call it, or how strange
>it looks.
I didn't say that the question looked strange, I said it looks strange
when you refer to yourself in the third person. I thought you
might like to know that.
It really does look very strange, in a megalomanic sort of way,
like you can't say "my question", it has to be "Eldon's Question",
because that's what it's going to be referred to in the history
books once the story about how mathematicians have all
been lying and Eldon has been the lonely voice in the
wilderness brave enough to speak the Truth is revealed.
If you like to sound that way go for it.
[...]
>
>> You should _publish_ your work on this question!
>> I mean you're one of the foremost experts on the
>> problem, perhaps _the_ expert, according to that
>> "paper" you wrote with Gray. Usually the foremost
>> expert on something is willing to publish his
>> results.
>>
>On this question, I'll go point to point with you, and you can't win.
>The truth is on my side.
Huh? So why don't you _publish_ the truth somewhere?
You alone know the Truth and you're hiding it from the
world - that's not very nice.
>> >Yet, you refuse to even say what the definition is. I've asked you
>> >repeatedly, "What does it mean to say given at least one?" If you've
>> >answered, I haven't seen it yet.
>> >
>> >There is an answer.
>> >
>> >Eldon
>> >
>> >> You're better off restricting your replies to clever one-liners
>> >> like "dumb fuck". Because then it doesn't look like your
>> >> _trying_ to make sense - here it looks like you're trying to
>> >> prove something by stating that stated questions exist,
>> >> and it comes off utterly hilarious.
>> >>
>You're left with the "dumb fucks" because the truth is on my side.
>
>Do you fear the truth?
You know this stuff about how people who disagree with you
are afraid to admit they're wrong, are afraid of the Truth, etc,
is really quite insulting. I also wonder whether it has a place
in the Baez crackpot index...
>Eldon
>
>
>
>> >>
>> >>
>> >>
>> >> ******************
>> >>
>> >> David C. Ullrich
>>
>>
>> ******************
>>
>> David C. Ullrich
******************
David C. Ullrich
Eldon Moritz
Heads and tails are equally likely.
The problem statement is true.
There are two separate and distinct coin flip sequences.
With HH, "at least one is a heads" is true. Call this the heads
statement.
With TT, "at least one is a tails" is true. Call this the tails
statement.
With HT, or TH, either statement would be true.
The coins can be flipped with the two statements equally likely, or
with extreme prejudice toward either side.
To get a correct answer of 1/3, the coins must be flipped with extreme
prejudice toward 'our side'. Flip with extreme prejudice toward
'heads', the answer to the tails question is 1.
Flip with no prejudice, and the answer to our question, or my
question, or "Eldon's question" is 1/2.
To onus to communicating the prejudice is on the flipper.
Dr. Ullrich, you always answer the extreme prejudice question. You let
my question go begging. With your technique, you always get the wrong
answer to my question.
Eldon
Eldon Moritz
> In article <b7guq8$ljk$1...@wisteria.csv.warwick.ac.uk>,
> ma...@mimosa.csv.warwick.ac.uk () writes:
> >In article <349f5619.03041...@posting.google.com>,
> > elmo...@yahoo.com (Eldon Moritz) writes:
> >>ma...@mimosa.csv.warwick.ac.uk () wrote in message news:<b7cmob$k85$1...@wisteria.csv.warwick.ac.uk>...
> >>> In article <foni9vg3100914oii...@4ax.com>,
> >>> ull...@math.okstate.edu writes:
> >>> >On Sun, 13 Apr 2003 10:08:29 +0000 (UTC),
> >>> >ma...@mimosa.csv.warwick.ac.uk () wrote:
> >>> >
> >...
> >
> ...
>
> PS to last post
>
> Eldon:
>
> I meant to say also that there is another serious weakness in your unbiased
> model for the "at least one is a head" problem, in that it does not
> generalize in any obvious way to more complicated problems. How about:
>
that it's simple. I didn't promise that you couldn't write an
ambiguous question, or that with more coins, and more words it
couldn't get complicated.
> There are 6 balls, which are red, green or yellow with equal probability,
> and at least one ball is red. What is the probability that all non-red
> balls are green?
>
> What is your unbiased model for this question? You may come up with an
> answer, but to be consistent, you will have to demonstrate that your
> model is the only possible correct one, and that any other model is just
> wrong. I can think of at least two possibilities which would give different
> answers.
>
colors, or were the colors selected, then the balls. As the problem is
written, It looks as though you started with equal numbers of red
green and yellow balls, then selected 6. Your statement, "at least one
is red", has to have been made after inspection. From here, the math
is a bit simpler.
> I am not attacking your model in particular, I am only disputing your claim
> that it is the only possible correct model.
>
> Derek Holt.
With two coin flips, they can be made with, or without prejudice.
Either the coins were flipped, and a head showed up, or, heads were
chosen, then flipped for. The onus is on the flipper to convey the
information.
When they were flipped without prejudice, the correct answer is 1/2.
The flipper told us of no prejudice. As written, the answer is 1/2.
As intended? Your guess is as good as mine.
We speak with linguistics, we argue with linguistics, we argue with
logic.
On this little question, our problem is not with the linguistics, we
have a logics problem. There is a difference in our logic. On this
question, my logic is correct.
Eldon
ma...@mimosa.csv.warwick.ac.uk
elmo...@yahoo.com (Eldon Moritz) writes:
>With two coin flips, they can be made with, or without prejudice.
>Either the coins were flipped, and a head showed up, or, heads were
>chosen, then flipped for. The onus is on the flipper to convey the
>information.
>
>When they were flipped without prejudice, the correct answer is 1/2.
>
>The flipper told us of no prejudice. As written, the answer is 1/2.
OK, suppose that we accept your interpretation that there really was a
flipper and he really is reporting on the event.
The flipper told us of no prejudice is not the same thing as the flipper
told us that there was no prejudice. I agree and have always agreed that
if there was no heads/tails prejudice involved in the flipper's report,
then the answer is 1/2. But you have no way of knowing whether or not there
was prejudice. Perhaps there was prejudice - you just cannot be certain!
So you cannot say for certain that the answer is 1/2.
As I keep repeating, if there is more than one interpretation of a problem
which is consistent with the information given in the problem, and the
two interpretations lead to different answers, then that problem is
ambiguous, and does not have a unqiue correct answer. You are welcome to
prefer your own interpretation, but you have not in any way proved that a
different interpretation is wrong.
Derek Holt.
Eldon Moritz
> In article <349f5619.03041...@posting.google.com>,
> elmo...@yahoo.com (Eldon Moritz) writes:
> >With two coin flips, they can be made with, or without prejudice.
> >Either the coins were flipped, and a head showed up, or, heads were
> >chosen, then flipped for. The onus is on the flipper to convey the
> >information.
> >
> >When they were flipped without prejudice, the correct answer is 1/2.
> >
> >The flipper told us of no prejudice. As written, the answer is 1/2.
>
> OK, suppose that we accept your interpretation that there really was a
> flipper and he really is reporting on the event.
>
was made by someone, or something. We have a right to assume that
heads and tails are equally likely. We have a right to assume that the
problem statement is true.
The statement tells us of 'a' coin flip. They were flipped by someone,
or something. I'll accept computer flips, imaginary flips, but there
was a flip; there was a flipper.
> The flipper told us of no prejudice is not the same thing as the flipper
> told us that there was no prejudice. I agree and have always agreed that
> if there was no heads/tails prejudice involved in the flipper's report,
> then the answer is 1/2. But you have no way of knowing whether or not there
> was prejudice. Perhaps there was prejudice - you just cannot be certain!
> So you cannot say for certain that the answer is 1/2.
>
I agree that "told us of no prejudice is not the same thing as the
flipper told us that there was no prejudice". However, when there was
no prejudice and the flipper told us nothing, is different again,
from, there was a prejudice and the flipper told us nothing. It falls
under the true statement argument.
Also:
We all agreed, even Ullrich, that the heads question, and the tails
question should have the same answer.
Suppose that we knew for certain that there was a prejudice, but we
didn't know which way. It's equally likely to have been a prejudice
toward heads as tails. As the answer for the heads question moves
toward 1/3, the answer for the tails question moves toward 1. This is
in violation of our earlier agreement.
Suppose that we know there was extreme prejudice, but we don't know
which way it was. (we certainly don't) We don't know if the flipper
was flipping for a head and found one, or flipping for a tail, and
didn't. That works our answer out to 1/2.
> As I keep repeating, if there is more than one interpretation of a problem
> which is consistent with the information given in the problem, and the
> two interpretations lead to different answers, then that problem is
> ambiguous, and does not have a unqiue correct answer. You are welcome to
> prefer your own interpretation, but you have not in any way proved that a
> different interpretation is wrong.
>
And I keep repeating, the information in the problem isn't consistent
with your interpretations, so far.
To get 1/3, the flipper had to be flipping for heads. As Martin
Gardner said, the flipper must agree in advance. As we have a written
question, this agreement would have to be a written agreement, written
right into the problem statement, such as "two coins were flipped
until at least one is a head." That 'until' is written agreement. The
poser of the question may have intended to put that 'until' in there;
just omitted it. We have no way of knowing that.
The question, as written, has correct answer 1/2. If there are
interpretations with correct answer 1/3, I haven't seen them yet.
There is a lot of mental reluctance around this quesion. Notice that
the illustrious Professor Ullrich, hasn't admitted, as of yet, that
the question is ambiguous. Most of my attackers are full fledged one
thirders. Once in a while, when they back into a corner, they say,
"well, maybe it's ambiguous." Most of them run off there. Once in a
while someone follows on through. Hopefully you have.
With the one thirder argument, how would they ever get the correct
answer to my question. Put the 'until' into the question, and we all
get the correct answer to their's.
Eldon
> Derek Holt.
Eldon Moritz
> On 12 Apr 2003 13:03:59 -0700, elmo...@yahoo.com (Eldon Moritz)
> wrote:
>
> >rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03041...@posting.google.com>...
> >> elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> >> > rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
> >> > > elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> >> > > > Two coins flip four ways. To flip two coins three ways, the
> >> > > > heads/tails decision must be made prior to the flip,
> >> > >
> >> > > Who said anything about "flipping two coins three ways"?
> >> > >
> >
> >This email articulates the difference between what I'm saying, and
> >what you think I'm saying.
> >
Eldon said the above.
> >What I'm actually saying is that I get strange answers to these
> >questions because I simply don't understand the meaning of
> >perfectly standard English constructions.
>
I don't believe Eldon said that. I researched the previous email
thoroughly. The above statement seems to have been 'manufactured' in.
Shakespearean scholar, and syndicated columnist, Joe Sobran, said that
when learned people get caught on the naive side, where they had to
rearrange a lot of mental furniture to keep up with a new thought.
They would manufacture, or make up stuff, in order to maintain their
incorrect position.
Methinks the learned professor has done this.
Eldon
> Ah. Yes, we have agreement on that.
>
> [...]
David C. Ullrich
wrote:
>David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<veug9vc6n38o2rgqo...@4ax.com>...
>> On 12 Apr 2003 13:03:59 -0700, elmo...@yahoo.com (Eldon Moritz)
>> wrote:
>>
>> >rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03041...@posting.google.com>...
>> >> elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
>> >> > rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
>> >> > > elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
>> >> > > > Two coins flip four ways. To flip two coins three ways, the
>> >> > > > heads/tails decision must be made prior to the flip,
>> >> > >
>> >> > > Who said anything about "flipping two coins three ways"?
>> >> > >
>> >
>> >This email articulates the difference between what I'm saying, and
>> >what you think I'm saying.
>> >
>
>Eldon said the above.
>
>> >What I'm actually saying is that I get strange answers to these
>> >questions because I simply don't understand the meaning of
>> >perfectly standard English constructions.
>>
>
>I don't believe Eldon said that. I researched the previous email
>thoroughly. The above statement seems to have been 'manufactured' in.
Really? Have you forgotten the fact that shortly before this post
_you_ started attributing things to _me_ that I never wrote?
I thought so.
>So, having shown that not only am I unable to understand simple
>English, I'm also unable to keep track of who said what when, I
>think I'll just shut up now until I have something new to say.
Good idea!
>
>Eldon
>
>
>> Ah. Yes, we have agreement on that.
>>
>> [...]
>> >
>> >Eldon
>>
>>
>> ******************
>>
>> David C. Ullrich
******************
David C. Ullrich
Eldon Moritz
> On 18 Apr 2003 05:30:48 -0700, elmo...@yahoo.com (Eldon Moritz)
> wrote:
>
> >David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<veug9vc6n38o2rgqo...@4ax.com>...
> >> On 12 Apr 2003 13:03:59 -0700, elmo...@yahoo.com (Eldon Moritz)
> >> wrote:
> >>
> >> >rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03041...@posting.google.com>...
> >> >> elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> >> >> > rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
> >> >> > > elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
> >> >> > > > Two coins flip four ways. To flip two coins three ways, the
> >> >> > > > heads/tails decision must be made prior to the flip,
> >> >> > >
> >> >> > > Who said anything about "flipping two coins three ways"?
> >> >> > >
> >> >
> >> >This email articulates the difference between what I'm saying, and
> >> >what you think I'm saying.
> >> >
> >
> >Eldon said the above.
> >
> >> >What I'm actually saying is that I get strange answers to these
> >> >questions because I simply don't understand the meaning of
> >> >perfectly standard English constructions.
> >>
> >
> >I don't believe Eldon said that. I researched the previous email
> >thoroughly. The above statement seems to have been 'manufactured' in.
>
> Really? Have you forgotten the fact that shortly before this post
> _you_ started attributing things to _me_ that I never wrote?
>
You really don't care about the correct answer for our question. Dr.
Holt seems to be getting it. Watch his posts carefully.
David C. Ullrich
wrote:
>David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<k1uv9vgu0d0ofnrof...@4ax.com>...
>> On 18 Apr 2003 05:30:48 -0700, elmo...@yahoo.com (Eldon Moritz)
>> wrote:
>>
>> >David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<veug9vc6n38o2rgqo...@4ax.com>...
>> >> On 12 Apr 2003 13:03:59 -0700, elmo...@yahoo.com (Eldon Moritz)
>> >> wrote:
>> >>
>> >> >rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03041...@posting.google.com>...
>> >> >> elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
>> >> >> > rpo...@yahoo.com (Randy Poe) wrote in message news:<585ab5d8.03040...@posting.google.com>...
>> >> >> > > elmo...@yahoo.com (Eldon Moritz) wrote in message news:<349f5619.03040...@posting.google.com>...
>> >> >> > > > Two coins flip four ways. To flip two coins three ways, the
>> >> >> > > > heads/tails decision must be made prior to the flip,
>> >> >> > >
>> >> >> > > Who said anything about "flipping two coins three ways"?
>> >> >> > >
>> >> >
>> >> >This email articulates the difference between what I'm saying, and
>> >> >what you think I'm saying.
>> >> >
>> >
>> >Eldon said the above.
>> >
>> >> >What I'm actually saying is that I get strange answers to these
>> >> >questions because I simply don't understand the meaning of
>> >> >perfectly standard English constructions.
>> >>
>> >
>> >I don't believe Eldon said that. I researched the previous email
>> >thoroughly. The above statement seems to have been 'manufactured' in.
>>
>> Really? Have you forgotten the fact that shortly before this post
>> _you_ started attributing things to _me_ that I never wrote?
>>
>I admit it - I'm guilty.
Good for you.
>
>You really don't care about the correct answer for our question. Dr.
>Holt seems to be getting it. Watch his posts carefully.
Guffaw. You don't seem to be _reading_ his posts.
>Eldon
>
>>
>>
>> ******************
>>
>> David C. Ullrich
******************
David C. Ullrich
ma...@mimosa.csv.warwick.ac.uk
elmo...@yahoo.com (Eldon Moritz) writes:
>David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<k1uv9vgu0d0ofnrof...@4ax.com>...
...
>> Really? Have you forgotten the fact that shortly before this post
>> _you_ started attributing things to _me_ that I never wrote?
>>
>This seems to be an admission of guilt.
>
>You really don't care about the correct answer for our question. Dr.
>Holt seems to be getting it. Watch his posts carefully.
No, I am not getting it, because as I have said ad nauseam, I do not
believe that there is a single correct answer to the question.
I differ from David Ullrich in that I have accepted that your solution is
a possible interpretation of the problem as written.
But there are at least two others. I still find David's belief that the
problem is purely abstract mathematical, and should be interpreted as being
identical to the same question with 'given' in it the most plausible.
But even if we agree that the problem is a report on an experiment that
actually took place, then there is still another possible interpretation
apart from yours, which is that the person reporting was biased. My
view on these things is that any interpreation which fits the facts
represents a possible solution, where some interpretations are more
plausible than others.
Derek Holt.
Eldon Moritz
> In article <349f5619.0304...@posting.google.com>,
> elmo...@yahoo.com (Eldon Moritz) writes:
> >David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<k1uv9vgu0d0ofnrof...@4ax.com>...
>
> ...
>
>
> >> Really? Have you forgotten the fact that shortly before this post
> >> _you_ started attributing things to _me_ that I never wrote?
> >>
> >This seems to be an admission of guilt.
> >
> >You really don't care about the correct answer for our question. Dr.
> >Holt seems to be getting it. Watch his posts carefully.
>
> No, I am not getting it, because as I have said ad nauseam, I do not
> believe that there is a single correct answer to the question.
>
when I'm wrong.
> I differ from David Ullrich in that I have accepted that your solution is
> a possible interpretation of the problem as written.
>
> But there are at least two others. I still find David's belief that the
> problem is purely abstract mathematical, and should be interpreted as being
> identical to the same question with 'given' in it the most plausible.
>
Back to my question, what does it mean to say we have and abstract
mathematical question, and how do we know? I've tried to show you that
that's wrong. It ain't a linguistics problem, it's a mathematical
error. The onus for the definitions is on the mathematician. The
definitions can't be just assumed into the question.
> But even if we agree that the problem is a report on an experiment that
> actually took place, then there is still another possible interpretation
> apart from yours, which is that the person reporting was biased. My
> view on these things is that any interpreation which fits the facts
> represents a possible solution, where some interpretations are more
> plausible than others.
>
We agreed that "at least one is a head" and "at least one is a tail"
is the same question, and should have the same answer.
The ONLY PLACE that happens, is with zero bias.
We can go either of those points, a point at a time. The second point,
about the bias, should be at rest with this post.
Hopefully, you're getting closer.
Eldon
> Derek Holt.
ma...@mimosa.csv.warwick.ac.uk
elmo...@yahoo.com (Eldon Moritz) writes:
>ma...@mimosa.csv.warwick.ac.uk () wrote in message news:<b7rjbd$7av$1...@wisteria.csv.warwick.ac.uk>...
...
>> I differ from David Ullrich in that I have accepted that your solution is
>> a possible interpretation of the problem as written.
>>
>> But there are at least two others. I still find David's belief that the
>> problem is purely abstract mathematical, and should be interpreted as being
>> identical to the same question with 'given' in it the most plausible.
>>
>Back to my question, what does it mean to say we have and abstract
>mathematical question, and how do we know? I've tried to show you that
>that's wrong. It ain't a linguistics problem, it's a mathematical
>error. The onus for the definitions is on the mathematician. The
>definitions can't be just assumed into the question.
OK, it's a matter of definition. Mathematical definitions are decided by
consensus, not by you alone!
>> But even if we agree that the problem is a report on an experiment that
>> actually took place, then there is still another possible interpretation
>> apart from yours, which is that the person reporting was biased. My
>> view on these things is that any interpreation which fits the facts
>> represents a possible solution, where some interpretations are more
>> plausible than others.
>>
>We agreed that "at least one is a head" and "at least one is a tail"
>is the same question, and should have the same answer.
This statement assumes that there is "an answer", which is exactly what I
am disputing. In fact many of your arguments have th hidden assumption
that the problem has to have a unique answer.
But I have just thought of yet another model, which has no heads/tails
bias and gives the answer 1/3 (I think).
Joe tosses the two coins, and then reports with a true statement of the
form "At least n coins are X", where n is 1 or 2, and X is either true or
false. He selects at random a true statement from the set of all possible true
statements, where all true statements have the same probability.
So, if he sees two heads, he says "At elast one is heads" or "At least
two are heads" with equal probability, and if he sees a head and a tail,
then he says "At least one is heads" or "At least one is tails" with equal
probability.
We know that he said "At least one is heads". What is the probability that
both are heads? I make that 1/3.
This is consistent with the given problem statement and displays no
heads/tails bias.
Derek Holt.
Eldon Moritz
> In article <349f5619.03041...@posting.google.com>,
> elmo...@yahoo.com (Eldon Moritz) writes:
> >ma...@mimosa.csv.warwick.ac.uk () wrote in message news:<b7rjbd$7av$1...@wisteria.csv.warwick.ac.uk>...
>
> ...
>
> >> I differ from David Ullrich in that I have accepted that your solution is
> >> a possible interpretation of the problem as written.
> >>
> >> But there are at least two others. I still find David's belief that the
> >> problem is purely abstract mathematical, and should be interpreted as being
> >> identical to the same question with 'given' in it the most plausible.
> >>
> >Back to my question, what does it mean to say we have and abstract
> >mathematical question, and how do we know? I've tried to show you that
> >that's wrong. It ain't a linguistics problem, it's a mathematical
> >error. The onus for the definitions is on the mathematician. The
> >definitions can't be just assumed into the question.
>
> OK, it's a matter of definition. Mathematical definitions are decided by
> consensus, not by you alone!
>
A mathematician can make a definition, for the scope of this paper.
In my arguments I have defined what you call bias, as "prior
prejudice". Dr. Gray has called it "extreme prejudice". Extreme
prejudice being the amount of prejudice needed to reach the answer
1/3.
We don't seem to disagree that the 'bias' is necessary. We don't seem
to disagree with what 'bias' is. Our disagreement is when we have it,
and when we don't.
We both agree on the definition of 'given at least one'. I have been
willing to put it into words. So far, you don't seem willing to do
this. When a student asks, "What does it mean to say, 'given at least
one?'". What do you tell said student?
You can tell student that with 'given at least one', the coins were
flipped with bias, or with extreme prejudice. Why? Because that's the
way it was defined in a special derivation of Bayes' theorem.
I win this point.
> >> But even if we agree that the problem is a report on an experiment that
> >> actually took place, then there is still another possible interpretation
> >> apart from yours, which is that the person reporting was biased. My
> >> view on these things is that any interpreation which fits the facts
> >> represents a possible solution, where some interpretations are more
> >> plausible than others.
> >>
> >We agreed that "at least one is a head" and "at least one is a tail"
> >is the same question, and should have the same answer.
>
> This statement assumes that there is "an answer", which is exactly what I
> am disputing. In fact many of your arguments have th hidden assumption
> that the problem has to have a unique answer.
>
When the coins were flipped with no bias, there is an answer and it's
1/2. Our flipper spoke of no bias.
Either, there was no bias, or,
our flipper declined to mention it.
In the first case, the flipper has the out, "I saw no bias, I had no
bias, I wanted no bias, I started no bias, I'm innocent."
In the second case, there was a bias, but the flipper mentioned no
bias. The flipper has no alibi. There is error in the works somewhere.
I win this point.
> But I have just thought of yet another model, which has no heads/tails
> bias and gives the answer 1/3 (I think).
>
> Joe tosses the two coins, and then reports with a true statement of the
> form "At least n coins are X", where n is 1 or 2, and X is either true or
> false. He selects at random a true statement from the set of all possible true
> statements, where all true statements have the same probability.
> So, if he sees two heads, he says "At elast one is heads" or "At least
> two are heads" with equal probability, and if he sees a head and a tail,
> then he says "At least one is heads" or "At least one is tails" with equal
> probability.
>
> We know that he said "At least one is heads". What is the probability that
> both are heads? I make that 1/3.
>
> This is consistent with the given problem statement and displays no
> heads/tails bias.
There is bias in your above example. You establish an extreme
prejudice toward heads.
Also, In our statement, there is absolutely no evidence, NONE, that
the flipper examined the second coin, therefore, your model has added
information to the statement.
You need more information to get the 1/3 answer, than is available in
our question.
I win this point.
I'll argue point to point, but I've covered most of these points
thoroughly in these two threads. My arguments hold, unless you can
knock them out. So far, most have been making the same old points,
causing me to re iterate mine.
The ambiguities are in the arguments. Not in our question. My
hypothesis from the start was that many mathematicians see the stated
question, then go straight to the definition. Ullrich has admitted
that's what he does.
You have said that's the most plausible, even in spite of argument to
the contrary. He is content in his little Bayesian cocoon. There is a
question whereas the coins were flipped and our statement was made.
The correct answer to that question is 1/2.
Ullrich never gets the correct answer to that question. That ain't
plausible.
When you see the other 'interpretations', you are thinking of an
obscure question, which you see has answer of 1/3.
The questions with correct answer of 1/3, have a bias.
The questions with correct answer of 1/2, were flipped without bias.
That ain't ambiguous. That's a fact.
I win this point.
Eldon
>
> Derek Holt.
ma...@mimosa.csv.warwick.ac.uk
elmo...@yahoo.com (Eldon Moritz) writes:
>ma...@mimosa.csv.warwick.ac.uk () wrote in message news:<b7tt2n$omp$1...@wisteria.csv.warwick.ac.uk>...
>> In article <349f5619.03041...@posting.google.com>,
>> elmo...@yahoo.com (Eldon Moritz) writes:
>> >ma...@mimosa.csv.warwick.ac.uk () wrote in message news:<b7rjbd$7av$1...@wisteria.csv.warwick.ac.uk>...
>>
>> ...
...
>
>> But I have just thought of yet another model, which has no heads/tails
>> bias and gives the answer 1/3 (I think).
>>
>> Joe tosses the two coins, and then reports with a true statement of the
>> form "At least n coins are X", where n is 1 or 2, and X is either true or
>> false. He selects at random a true statement from the set of all possible true
>> statements, where all true statements have the same probability.
>> So, if he sees two heads, he says "At elast one is heads" or "At least
>> two are heads" with equal probability, and if he sees a head and a tail,
>> then he says "At least one is heads" or "At least one is tails" with equal
>> probability.
>>
>> We know that he said "At least one is heads". What is the probability that
>> both are heads? I make that 1/3.
>>
>> This is consistent with the given problem statement and displays no
>> heads/tails bias.
>
>There is bias in your above example. You establish an extreme
>prejudice toward heads.
Nonsense! The setup I described is completely symmetrical in heads/tails.
Where is the lack of symmetry?
On the contrary, your model is biased in favour of the number "one".
I have removed this bias, by allowing the possibility that the flipper
says "At least two are heads" or "at least two are tails".
>Also, In our statement, there is absolutely no evidence, NONE, that
>the flipper examined the second coin, therefore, your model has added
>information to the statement.
Now you are contradicting yourself.
According to *your* model, the flipper examines both coins, and if (s)he
sees heads/tails, (s)he says "At least one is heads" or "At least one
is tails" with equal probability.
What do you mean by "the second coin" anyway? The problem statement does not
introduce any ordering on the coins - it says "at least one is heads".
As soon as you start talking about first and second coins, it is you who
are adding information to the statement.
Derek Holt.