Me and David C. Ullrich
Elmo
at least one is a head. What are the chances that there are two heads?"
Many mathematicians get it confused with "The probability for two
heads, given at least one head?"
I say that "given at least one head", and told, "at least one head"
mean two different things.
Dr. Ullrich has stated in this forum, that the two mean the same thing.
If for no other reason than that he said so. Also because everyone
assumes them to be the same. (everyone does not, I don't)
I had occasion to visit Oklahoma State University. I found Dr. Ullrich
in the Student Center. He said I was not welcome there. He refused to
discuss the question with me there, as he has refused to discuss it
here.
The probability for two tails, given at least one tail is one third.
That is defined mathematically and is not arguable.
What it means to say, "at least one is a tail" is defined in the world
domain and means what it says. As the math domain is a subset of the
world domain, it is not possible to re-define, inside the math domain,
what "at least one is" means in the outside world.
An example:
Suppose that a computer program randomizes the coins, or dice, or puppy
dogs. Then shows us to colors. Suppose that we see a red and an orange.
We can make the statement"there are two colors and at least one is red"
or we can make the statement "there are two colors and at least one is
orange." The red statement tells nothing of orange, and the orange
statement tells nothing of red.
The statements "at least one is red" and "at least one is orange"
demonstrate what it means to say "at least one is".
Now make the statement, "There are two colors, given at least one red."
Does that mean the same thing as "There are two coins, given at least
one head"? If it does, then the probability for two reds is 1/3. To
mean the same thing as "given at least one", the 'red' statement must
convey some knowledge of 'orange' and the 'orange' statement must
convey some knowledge of 'red'.
Dr. Ullrich seems to be somewhat of a fakir. He plays a lot of
bachgammon.
He argues a lot in this forum.
He does not seem to publish.
He draws a paycheck at OSU, I guess, wonder if he has tenure?
When you accept that "given at least one" and told, "at least one" are
different; then it's easy to get from:
P(hh|at least one h) = 1/3 to
P(hh|told "at least one h) = 1/2.
Eldon Moritz
elmo...@yahoo.com
José Carlos Santos
> He does not seem to publish.
According to the MR Lookup service:
David Ullrich published 28 papers. Of course, there's the possibility
that there are other Mathematicians called David Ullrich, but *some*
of these articles were actually written by David C. Ullrich.
Oddly, the only article that I was able to locate written by a "E.
Moritz" is an article written in german and published in 1948. I doubt
that it something to do with you.
Best regards,
Jose Carlos Santos
Roger Bagula
Now, you have met him!
Dave L. Renfro
>> He [David C. Ullrich] does not seem to publish.
José Carlos Santos wrote:
> According to the MR Lookup service:
>
> http://www.ams.org/mrlookup
>
> David Ullrich published 28 papers. Of course,
> there's the possibility that there are other
> Mathematicians called David Ullrich, but *some*
> of these articles were actually written by
> David C. Ullrich.
>
> Oddly, the only article that I was able to locate
> written by a "E. Moritz" is an article written in
> german and published in 1948. I doubt that it
> something to do with you.
Here are a couple of papers that you missed,
and they're in English. However, their existence
doesn't weaken your argument, nor my belief
in the correctness of what you were implying.
http://www.emis.de/cgi-bin/JFM-item?33.0303.01
http://www.emis.de/cgi-bin/JFM-item?46.0915.01
Dave L. Renfro
The Idiot's Menace
news:1128780755.1...@g44g2000cwa.googlegroups.com...
[snip]
> I had occasion to visit Oklahoma State University. I found Dr. Ullrich
> in the Student Center. He said I was not welcome there. He refused to
> discuss the question with me there, as he has refused to discuss it
> here.
[snip]
> Dr. Ullrich seems to be somewhat of a fakir. He plays a lot of
> bachgammon.
> He argues a lot in this forum.
> He does not seem to publish.
> He draws a paycheck at OSU, I guess, wonder if he has tenure?
[snip]
He is also an asshole. If this isn't as obvious to you as the fact that the
sun rises every day, you are blind.
If it was only up to him to teach new/prospective students, mathematics
would be a dead science by now. Neither does he care about it, either.
You are wasting your time arguing with him.
> Eldon Moritz
> elmo...@yahoo.com
José Carlos Santos
> Here are a couple of papers that you missed,
> and they're in English. However, their existence
> doesn't weaken your argument, nor my belief
> in the correctness of what you were implying.
Mainly because one of them was published in 1917 and the other one in
1901... :-)
Jim Spriggs
> I had occasion to visit Oklahoma State University. I found Dr. Ullrich
> in the Student Center. He said I was not welcome there.
Do you have any right to be in the OSU Student Center?
--
I don't know who you are Sir, or where you come from,
but you've done me a power of good.
Elmo
Jim Spriggs wrote:
> Elmo wrote:
>
> > I had occasion to visit Oklahoma State University. I found Dr. Ullrich
> > in the Student Center. He said I was not welcome there.
>
> Do you have any right to be in the OSU Student Center?
>
Eldon
NotP
>
> Jim Spriggs wrote:
>> Elmo wrote:
>>
>> > I had occasion to visit Oklahoma State University. I found Dr. Ullrich
>> > in the Student Center. He said I was not welcome there.
>>
>> Do you have any right to be in the OSU Student Center?
>>
> I bought an OSU cap, and lunch. They took my money.
> Eldon
>
Accosting someone you don't know like that is creepy and uncalled for. This
is one of the scary things about giving your identity on the internet.
Jim Spriggs
>
> Jim Spriggs wrote:
> > Elmo wrote:
> >
> > > I had occasion to visit Oklahoma State University. I found Dr. Ullrich
> > > in the Student Center. He said I was not welcome there.
> >
> > Do you have any right to be in the OSU Student Center?
> >
> I bought an OSU cap, and lunch. They took my money.
> Eldon
Fairy snuff, if they took your money. But does that entitle you to
harass David C Ullrich?
mensa...@aol.com
Elmo wrote:
> On this forum I have argued the question, "Two coins were flipped and
> at least one is a head. What are the chances that there are two heads?"
>
> Many mathematicians get it confused with "The probability for two
> heads, given at least one head?"
>
> I say that "given at least one head", and told, "at least one head"
> mean two different things.
Really?
>
> Dr. Ullrich has stated in this forum, that the two mean the same thing.
> If for no other reason than that he said so. Also because everyone
> assumes them to be the same. (everyone does not, I don't)
Well, mathematics is not a democracy. But it isn't a
parlimentary system either. There are no minority opinions
allowed, only Truth. And if you think your opinion is Truth,
then you have two choices:
- prove you are right
or
- prove the majority is wrong
>
> I had occasion to visit Oklahoma State University. I found Dr. Ullrich
> in the Student Center. He said I was not welcome there. He refused to
> discuss the question with me there, as he has refused to discuss it
> here.
Here's a probability problem for you: what do you think the
odds are, given this stalking incident and the evidence in these
sci.math postings, that a judge would sign a restraining order
against you if Dr. Ullrich requested it?
>
> The probability for two tails, given at least one tail is one third.
> That is defined mathematically and is not arguable.
Then why are you arguing?
>
> What it means to say, "at least one is a tail" is defined in the world
> domain and means what it says. As the math domain is a subset of the
> world domain, it is not possible to re-define, inside the math domain,
> what "at least one is" means in the outside world.
Huh?
>
> An example:
>
> Suppose that a computer program randomizes the coins, or dice, or puppy
> dogs. Then shows us to colors. Suppose that we see a red and an orange.
>
> We can make the statement"there are two colors and at least one is red"
> or we can make the statement "there are two colors and at least one is
> orange." The red statement tells nothing of orange, and the orange
> statement tells nothing of red.
Ok.
>
> The statements "at least one is red" and "at least one is orange"
Wait a minute. That's TWO statements.
> demonstrate what it means to say "at least one is".
You're trying to put ten pounds of shit into a five pound bag.
>
> Now make the statement, "There are two colors, given at least one red."
> Does that mean the same thing as "There are two coins, given at least
> one head"? If it does, then the probability for two reds is 1/3. To
> mean the same thing as "given at least one", the 'red' statement must
> convey some knowledge of 'orange' and the 'orange' statement must
> convey some knowledge of 'red'.
The statement "given at least one" does NOT specify WHICH one
is heads. If the two coins were a dime and a nickel, then
"given the nickel is heads" DOES convey information about the
dime and would make your reasoning correct. But it doesn't say
that, "given at least one" leaves the issue of WHICH coin is
heads unspecified and your reasoning does not apply in that case.
>
> Dr. Ullrich seems to be somewhat of a fakir. He plays a lot of
> bachgammon.
> He argues a lot in this forum.
> He does not seem to publish.
> He draws a paycheck at OSU, I guess, wonder if he has tenure?
>
> When you accept that "given at least one" and told, "at least one" are
> different;
False premise.
> then it's easy to get from:
>
> P(hh|at least one h) = 1/3 to
> P(hh|told "at least one h) = 1/2.
It matters not if your logical inferrences are valid, the false
premise renders your conclusion meaningless.
>
> Eldon Moritz
> elmo...@yahoo.com
Dave L. Renfro
> Mainly because one of them was published in 1917
> and the other one in 1901... :-)
Actually, I think I have a copy of the 1901 paper
around here somewhere, along with two or three others
that re-discover the same idea!
E. Moritz, "Quotientiation, an extension of the
differentiation process", Nebraska Acad. Proceed. S.
112-117. [JFM 33.0303.01]
http://www.emis.de/cgi-bin/JFM-item?33.0303.01
I'm not sure where my copy of Moritz's paper is
(i.e. which box of unsorted papers it's in), but
I do have a privately printed booklet on my bookshelf
(a bookcase in another room to be more correct, but
at least the stuff on it is easy to find and get at)
that re-discovers the same idea, namely the idea of
replacing "difference-quotient" with other operation
pairs, such as "quotient-root":
Michael Grossman and Robert Katz, "Non-Newtonian
Calculus", 7'th printing, Mathco (Box 240, Rockport,
Massachusetts) 1979, viii + 96 pages.
Hummm...Apparently, this pamphlet isn't as unknown as
I thought.
http://www.google.com/search?as_epq=non+newtonian+calculus
http://groups.google.com/groups?as_epq=non+newtonian+calculus
Besides Moritz's paper above, I found two other papers
mentioned on-line that deal with this method of
differentiation, and maybe they are the papers I was
thinking of, although I'm pretty sure I also recall
seeing something like this in one or two German papers
that appeared between 1880 and 1900.
http://www.emis.de/projects/EULER/search?q=Quotientiation
Dave L. Renfro
quasi
>On this forum I have argued the question, "Two coins were flipped and
>at least one is a head. What are the chances that there are two heads?"
>
>Many mathematicians get it confused with "The probability for two
>heads, given at least one head?"
>
>I say that "given at least one head", and told, "at least one head"
>mean two different things.
>
> ...
>
>When you accept that "given at least one" and told, "at least one" are
>different; then it's easy to get from:
>
>P(hh|at least one h) = 1/3 to
>P(hh|told "at least one h) = 1/2.
>
If information is "given", how is it given? Someone tells you, right?
I can't think of any other way to interpret "given".
For example, consider the following 2 problems:
(1) Given x=3. What is the value of x+1?.
(2) You are told x=3. What is the value of x+1?
I'm certain that most people would not see any difference in the above
problems.
As another example, let's take your example but with just 1 coin.
Consider the following 2 problems:
(1) One coin is flipped. Given that there is at least one head, what
is the probability that it is a head?
(2) One coin is flipped. If you are told that there is at least one
head, what is the probability that it is a head?
It's obvious to you, I hope, that the answer to both questions is 1.
Ok, so again, no difference.
Now for the problem you raised with 2 coins, you seem to think that
being "given" some info and being "told" the same info is somehow
different, and as a result, we get 2 different answers for the
following problem:
2 coins are flipped. You are told that there is at least one head.
What is the probability of 2 heads
You think the answer is 1/2 whereas my answer is 1/3.
To bring you back to reality, I'll propose a bet based on the
following experiment.
A referee flips 2 coins, but flipped coins are hidden from public
view. If both are tails, the referee reflips both coins. If at least
one is a head then the referee announces:
"There is at least one head."
If you think the probability of 2 heads is 1/2, then presumably you
would be willing to bet even money on 2 heads. After all, you view it
as 50-50, right?
Well, if you really believe it's 50-50 then quick, go get some money,
as much as you can afford to lose, and let's go get a ref and a fair
coin. You bet on 2 heads, I'll bet against.
To make the bet more attractive, instead of even money, I'll offer you
3-2 odds. In other words, if the result is 2 heads, you win and I'll
pay you 3, otherwise you lose, but you only have to pay me 2.
If you agree to play this game, one of us is sure to go broke. Who do
you think that would be?
quasi
Richard Tobin
Elmo <elmo...@yahoo.com> wrote:
>On this forum I have argued the question, "Two coins were flipped and
>at least one is a head. What are the chances that there are two heads?"
The obvious interpretation of this is that you are restricting the
cases to those in which at least one is a head, and asking what the
probability of two heads is in this case.
If you mean something else, you should expect to be misunderstood
often.
>I had occasion to visit Oklahoma State University. I found Dr. Ullrich
>in the Student Center. He said I was not welcome there. He refused to
>discuss the question with me there, as he has refused to discuss it
>here.
Bad luck, you could probably have learnt a lot from him.
-- Richard
Robert Low
> If you agree to play this game, one of us is sure to go broke. Who do
Technically, isn't that 'almost sure' to go broke?
john_r...@sagitta-ps.com
Elmo wrote:
>
> On this forum I have argued the question, "Two coins were flipped
> and at least one is a head. What are the chances that there are
> two heads?"
>
> Many mathematicians get it confused with "The probability for two
> heads, given at least one head?"
>
> I say that "given at least one head", and told, "at least one head"
> mean two different things.
>
> Dr. Ullrich has stated in this forum, that the two mean the same
> thing. If for no other reason than that he said so. Also because
> everyone assumes them to be the same. (everyone does not, I don't)
I can safely say that everyone who speaks English in the whole
world, except you, would agree they mean exactly the same.
> I had occasion to visit Oklahoma State University. I found
> Dr. Ullrich in the Student Center. He said I was not welcome
> there. He refused to discuss the question with me there, as
> he has refused to discuss it here.
Unlike Dr Ullrich I'd be more than willing to discuss it
in person, if you can make your way to the University of
Ulan Bator in Mongolia, where I teach maths.
Tell you what, I'll even buy you lunch; they do an excellent
goat casserole here.
Best Wishes
Dr J Ramsden
(Maths Dept, Ulan Bator University)
John O'Flaherty
>
> When you accept that "given at least one" and told, "at least one" are
> different; then it's easy to get from:
>
> P(hh|at least one h) = 1/3 to
> P(hh|told "at least one h) = 1/2.
If you say 'coin 1 is heads, what is the chance coin 2 is heads'?',
then the chance is 1/2, because the case you are looking for is H1H2,
when the possible outcomes are H1H2 and H1T2.
If you say 'one coin is heads, what is the chance both coins are
heads?, then the chance of H1H2 is 1/3, because the possible outcomes
are H1T1, T1H2 and H1H2. The outcome of focus is one of three equally
probable outcomes.
Conditional probability is
P[A|B] = P[AB]/P[B].
A = {H1 and H2} for both cases, with P[A] = 1/4
P[AB] is the same as P[A]- the probability of {A and AB}.
For the first case, B = H1, with P[B] = 1/2
so P[A|B] = .25/.5 = 1/2
For the second case, "given at least one head" is equivalent to "being
told there is at least one head". It is B = {H2 or H1 or (H1 and H2)},
with P[B] = (1/2 + 1/2 - 1/4) = 3/4. Then, P[AB] is still equal to
P[A], or 1/4, and 1/4 divided by 3/4 is 1/3.
--
john
Richard Tobin
<john_r...@sagitta-ps.com> wrote:
>I can safely say that everyone who speaks English in the whole
>world, except you, would agree they mean exactly the same.
I think that's an exaggeration. The expression "probability of A
given B" is not an expression that everyone is familiar with. A few
years ago there was a series of letters in one of the national
newspapers here about this problem, and it was clear that not
everyone took it the intended way.
The problem with the OP is not his initial misunderstanding, but
his attitude to someone who explained things to him.
-- Richard
mensa...@aol.com
With camel butter?
Joubin Houshyar
The probability **for the person throwing the coins**, the ref, and
producing 2 heads is, in fact, 1/3.
Those who argue for 1/2 and make a semantic case regarding "given" are,
in effect, imo, arguing for the subjective decoupling/decomposition of
the system 'state' from the system 'dynamics':
In isolation:
If you are told there is are two coins on a table and "given"
information that one of them is in fact a "head", then what are the
possibilities for the other coin? In isolation, one can reasonably
argue that it is either heads or tails, thus (at that point) they have
a 1/2 probability of being right if they do claim it is both "heads".
However, should they choose (as you suggest) to continuously bet on the
process, then clearly they would be subject to the probability of the
process (and not just the probability of guessing a distinct, isolated,
state) and would lose johny's college tuition fund.
So, if you would allow me to 'intermittently' (and randomly :) show up
at the table, then I think 1/2 is actually a safe bet.
By NOT placing a bet for every sequence I *am* decoupled from the
probabilities inherent to the system 'dynamics'.
However, if you insist that my bets be sequentially congruent to your
system and don't let me to just randomly place bets on your table, then
YOU are correct as regards to 1/3 being the actual probability *for
me*.
/R
Joubin Houshyar
Joubin Houshyar wrote:
> By NOT placing a bet for every sequence I *am* decoupled from the
> probabilities inherent to the system 'dynamics'.
>
> However, if you insist that my bets be sequentially congruent to your
> system and don't let me to just randomly place bets on your table, then
> YOU are correct as regards to 1/3 being the actual probability *for
> me*.
I'll retract this. It will be 1/3, regardless.
/R
quasi
wrote:
>quasi wrote:
>> If you agree to play this game, one of us is sure to go broke. Who do
>
>Technically, isn't that 'almost sure' to go broke?
Hehe -- I'll settle for almost surely.
David C. Ullrich
>[...]
>I had occasion to visit Oklahoma State University. I found Dr. Ullrich
>in the Student Center. He said I was not welcome there.
I may have been wrong about that - I told you what I thought it
said on the door of the building, but later I couldn't find that
statement.
Anyway, as long as you feel like mentioning this behavior
of yours in public, let's give the full story:
I arrived at the Student Union looking for lunch. You introduced
yourself. I was politely dismissive and walked over to buy a
sandwich.
You joined me in line and offered to buy me lunch. I politely
declined, saying I'd rather buy my own lunch. You went back to
your table.
I got my sandwich. As I was looking for a table you invited
me to join you. I didn't, instead finding my own table.
You didn't seem to realize that if I wanted to discuss your
obsession I would have joined you - you got up and
joined me at my table, uninvited. You started going on
about heads and tails - I said I didn't want to discuss
it.
After a while I gave up - I got up and moved to
another table in an adjoining room. You followed me
and joined me at the second table, again uninvited,
in spite of the fact that it _must_ have been clear
to you that I was not interested in having you join
me for lunch.
I pointed out (erroneously, it appears) that on the
door of the building it said that the Student Union
was open to university students and staff, and to
the public for special events. I repeated this a
few times and you finally left.
A few days later I got an email from you explaining
that I should not have been intimidated by a little
old guy like you. Since I had nothing to say to that
(intimidation was not the problem) I didn't reply.
Some time after that I got another email from you,
in which you threatened to report me to the
university administration (I _think_ you specified
the president and the chancellor) if I continued
to refuse to discuss the question with you.
I replied to that one, explaining that while you
had the right to post whatever you want on the
internet you do _not_ have the right to intrude
on my personal life this way. You don't - the
next time you behave this way I _will_ be
calling the university police.
No, you're not welcome to join me at lunch
uninvited. No, you're not welcome in my office,
unless some day you're enrolled in a class I'm
teaching. No, you're not welcome to show up on
my door step at home.
No, I'm not required to explain why you're
not welcome in any of those places, any more
than I'm required to discuss the question
about heads and tails with you.
>He refused to
>discuss the question with me there, as he has refused to discuss it
>here.
The _reason_ you're free to post whatever you want on the internet,
as oppposed to harassing me the way you did that day, is that
when you post something on the internet anyone who feels like
paying no attention can simply ignore your post. But for
the record, it's not true that I've refused to discuss this
with you here. I've explained that your assertion that this
and that mean different things is simply wrong. And that's
all the discussion that _that_ question merits - if you
have a problem understanding simple English it seems
pointless to try to explain things to you, because
there's no reason to think that you're going to
understand the English in the explanation any better
than you understand that this and that do mean the
same thing.
>[...]
>
>Dr. Ullrich seems to be somewhat of a fakir. He plays a lot of
>bachgammon.
That's "faker", and "backgammon".
>He argues a lot in this forum.
>He does not seem to publish.
Hmm. That's a comment on my professional competence. And
it's demonstrably false, as others here have already
pointed out - it's very easy to go to standard public
references and _see_ that it's false.
You should really be careful about making false
statements about people in public. I'm free to
express my opinion that George Bush is an idiot, and
that appointing him president was one of the biggest
blunders the Supreme Court ever made. But if I
said that he seems to molest children in his spare
time I could get in trouble for that. Just a thought.
>He draws a paycheck at OSU, I guess, wonder if he has tenure?
>
>When you accept that "given at least one" and told, "at least one" are
>different; then it's easy to get from:
>
>P(hh|at least one h) = 1/3 to
>P(hh|told "at least one h) = 1/2.
>
>Eldon Moritz
>elmo...@yahoo.com
************************
David C. Ullrich
Jesse F. Hughes
> "Elmo" <elmo...@yahoo.com> wrote in message
> news:1128796258....@z14g2000cwz.googlegroups.com...
>>
>> Jim Spriggs wrote:
>>> Elmo wrote:
>>>
>>> > I had occasion to visit Oklahoma State University. I found Dr. Ullrich
>>> > in the Student Center. He said I was not welcome there.
>>>
[...]
>
> Accosting someone you don't know like that is creepy and uncalled
> for.
Absolutely.
If Elmo wanted to chat with Ullrich, he should have called and asked
for an appointment. Surprising Ullrich in a public place to argue
with him is just plain creepy.
--
"Yup, you guessed it. If worse comes to worse, I *will* turn to the
Army to help me with mathematicians. And then mathematicians don't
think the NSA or CIA can save your asses, as generals LIKE me."
-- James Harris's latest foray into mathematical logic.
James Dolan
x given that someone tells you that y?" can (of course) have quite
different answers (and other similar phrasings can have further
different answers). ordinary language is forever blurring subtle
distinctions like this, but it's even worse to see the mob mindlessly
insisting that the distinction doesn't exist than it would be to see
someone insisting that ordinary language is unambiguous enough that we
should all be able to agree about when the distinction is intended.
--
[e-mail address jdo...@math.ucr.edu]
john_r...@sagitta-ps.com
James Dolan wrote:
>
> ordinary language is forever blurring subtle distinctions
> like this, but it's even worse to see the mob mindlessly
> insisting that the distinction doesn't exist than it
> would be to see someone insisting that ordinary language
> is unambiguous enough that we should all be able to agree
> about when the distinction is intended.
I agree it's wrong to blur subtle distinctions; but where
is the distinction between the following statements?
"Two coins were flipped and at least one is a head.
What are the chances that there are two heads?"
"Two coins were flipped. What are the chances that
there are two heads, given that at least one is a head?"
It could be argued that the first version states directly
that at least one is a head, whereas in the second this
is conveyed by a meta-statement that this fact has been
reported or somehow indicated.
But within the confines of the questions, we must assume
that the relevant information and conditions are equally
accurate. So for all practical purposes they differ only
in syntax.
The way I see it, this dispute isn't about probability
but only initial grammatical interpretation (and to argue
otherwise is to blur the distinction between these ;-)
I still maintain that to anyone with an adequate grasp
of English and familiar with expressing a condition by
means of a "given" clause (whose use isn't confined to
maths BTW), the two sentences mean exactly the same.
Cheers
Dr John R Ramsden (j...@ulanbator.edu.com)
David Ames
I was about to assume that Ulan Bator University's name did not change
when the official spelling of "Ulan Bator" changed. But then I
attempted to reach ulanbator.edu.com and was unable to do so. I wonder
if anyone has a Domain Name Server that allows ulanbator.edu.com to be
reached? I am puzzled.
David Ames
john_r...@sagitta-ps.com
quasi wrote:
>
> On 8 Oct 2005 07:12:35 -0700, "Elmo" <elmo...@yahoo.com> wrote:
>
> >On this forum I have argued the question, "Two coins were flipped and
> >at least one is a head. What are the chances that there are two heads?"
> >
> >Many mathematicians get it confused with "The probability for two
> >heads, given at least one head?"
> >
> >I say that "given at least one head", and told, "at least one head"
> >mean two different things.
> >
> > ...
> >
> >When you accept that "given at least one" and told, "at least one" are
> >different; then it's easy to get from:
> >
> >P(hh|at least one h) = 1/3 to
> >P(hh|told "at least one h) = 1/2.
> >
>
> If information is "given", how is it given? Someone tells you,
> right? I can't think of any other way to interpret "given".
Exactly, and the natural interpretation is that this condition
is being given by the person stating the question. Even if not,
one must assume that the information is equally accurate. So,
although there may be a hairsplitting logical distinction in
how the question is framed, the salient facts are the same.
If the condition had been given by someone else, who tells lies
half the time or something, then yes it would be an different
question. But surely everyone agrees that a problem like this
must be taken how one finds it, as an abstract exercise, without
trying to read between lines and perversely imagine all kinds of
complicating factors, even if it might be appropriate to do that
in real life.
john_r...@sagitta-ps.com
David Ames wrote:
>
> John Ramsden wrote:
> >
> > I still maintain that to anyone with an adequate grasp
> > of English and familiar with expressing a condition by
> > means of a "given" clause (whose use isn't confined to
("exactly" changed to "essentially")
> > Cheers
> >
> > Dr John R Ramsden (j...@ulanbator.edu.com)
>
> I was about to assume that Ulan Bator University's name did
> not change when the official spelling of "Ulan Bator" changed.
> But then I attempted to reach ulanbator.edu.com and was unable
> to do so. I wonder if anyone has a Domain Name Server that
> allows ulanbator.edu.com to be reached? I am puzzled.
Ah, that's a spam trap - we get the wretched stuff even here.
I shouldn't worry about it. We'll only confuse Elmo even more.
(He's probably booking his flight even as we speak - see my
first post in this thread.)
Elmo
I'm 69 years old, and skinny. I'm not a physical threat. I was in his
territory, surrounded by his people. I did not haraass him.
Eldon
James Dolan
<john_r...@sagitta-ps.com> wrote:
sorry, your question is just too stupid to bother answering. if you'd
really like an answer then i suggest that you read what i wrote once
more and this time don't hallucinate that i said things completely
other than what i actually said.
--
[e-mail address jdo...@math.ucr.edu]
Elmo
Two coins were flipped. Given that they both landed tails, and the
statement was generated, "Two coins were flipped and at least one
landed tails. What are the chances for two tails?"
Bruce only heard the statement, he bet one token for two tails. Bruce
will win. What odds should he collect?
Or:
Given that they both landed heads and the statement was generated, "Two
coins were flipped and at least one landed heads. What are the chances
for two heads?" Bruce only heard the statement, and bet for two heads.
He will win. What odds should he collect?
Or: Given that they landed one of each and either statement was
generated. Bruce will lose his token, the odds are moot.
When Bruce wins, if the answer to the question is 1/3, he should get
his token back, plus two.
Here, it's given that Bruce was told, "at least one." This is
definitely different from being given at least one.
Eldon:)
David C. Ullrich
>Harass? He said I wasn't welcome. I left.
Guffaw. Yes, you left _eventually_.
>I'm 69 years old, and skinny. I'm not a physical threat. I was in his
>territory, surrounded by his people. I did not haraass him.
Nobody said anything about a physical threat.
>Eldon
************************
David C. Ullrich
Christian Bau
john_r...@sagitta-ps.com wrote:
> quasi wrote:
> >
> > If information is "given", how is it given? Someone tells you,
> > right? I can't think of any other way to interpret "given".
>
> Exactly, and the natural interpretation is that this condition
> is being given by the person stating the question. Even if not,
> one must assume that the information is equally accurate. So,
> although there may be a hairsplitting logical distinction in
> how the question is framed, the salient facts are the same.
Lets say two coins have been thrown, I can't see the result, but you
can. I want to find out a bit more. I ask a question and you have to
answer thruthfully.
Case 1: I ask "Is at least one coin heads?", and the only possible
statements that you can make are "None of the coins is heads" and "at
least one of the coins is heads". This is the same as "... given that
..."
Case 2: I tell you: "Please give me some true statement about the
coins". Now you have an infinite amount of choices. This is the same as
"... somebody told me that..."
Case 3: I tell you: "Please give me some true statement about the coins.
I'll give you one dollar if you say that at least one coin is heads.
I'll give you one thousand dollars if you say both coins are heads. If
you say anything else, or if you lie, I'll give you nothing. "
David C. Ullrich
jdo...@math-cl-n03.math.ucr.edu (James Dolan) wrote:
>"what's the probability of x given y?" and "what's the probability of
>x given that someone tells you that y?" can (of course) have quite
>different answers
Exactly what is the difference in meaning between
"Two coins were flipped and at least one is a head. What
are the chances that there are two heads?"
and
"The probability for two heads, given at least one head?"
?
>(and other similar phrasings can have further
>different answers). ordinary language is forever blurring subtle
>distinctions like this, but it's even worse to see the mob mindlessly
>insisting that the distinction doesn't exist
I don't recall anyone insisting that there is no difference
between "what's the probability of x given y?" and
"what's the probability of x given that someone tells you that y?"
>than it would be to see
>someone insisting that ordinary language is unambiguous enough that we
>should all be able to agree about when the distinction is intended.
************************
David C. Ullrich
Victor Eijkhout
> >I'm 69 years old, and skinny. I'm not a physical threat. I was in his
> >territory, surrounded by his people. I did not haraass him.
>
> Nobody said anything about a physical threat.
Surely you're not implying he was an intellectual threat?
Victor.
--
email: lastname at cs utk edu
homepage: www cs utk edu tilde lastname
Dave L. Renfro
>> Nobody said anything about a physical threat.
Victor Eijkhout wrote:
> Surely you're not implying he was an intellectual threat?
Some possibilities:
A threat to his brief time away from students asking
him things like what they need to get on the next test
to have a 'C' average (a high school algebra 1 problem
that many college students can't solve), why they got
8 points off of a 10 point problem where they were to
evaluate the derivative of (x^2)*sin(x) and they put
(2x)*cos(x) ("I got all the derivatives correct, and
you're taking off _8_ points just because I didn't
write them down exactly like you wanted us to?!?"), etc.
A threat to his being able to think over what he plans
to cover in a class that meets an hour from then.
A threat to the time he time he intended to spend
with the girlfriend his wife doesn't know about.
(Ooops--Sorry about this slip, David!)
Dave L. Renfro
john_r...@sagitta-ps.com
Christian Bau wrote:
>
> In article <1128860336.2...@f14g2000cwb.googlegroups.com>,
> john_r...@sagitta-ps.com wrote:
>
> > quasi wrote:
> > >
> > > If information is "given", how is it given? Someone tells you,
> > > right? I can't think of any other way to interpret "given".
> >
> > Exactly, and the natural interpretation is that this condition
> > is being given by the person stating the question. Even if not,
> > one must assume that the information is equally accurate. So,
> > although there may be a hairsplitting logical distinction in
> > how the question is framed, the salient facts are the same.
>
> Lets say two coins have been thrown, I can't see the result, but you
> can. I want to find out a bit more. I ask a question and you have to
> answer thruthfully.
>
> Case 1: I ask "Is at least one coin heads?", and the only possible
> statements that you can make are "None of the coins is heads" and "at
> least one of the coins is heads". This is the same as "... given that
> ..."
>
> Case 2: I tell you: "Please give me some true statement about the
> coins". Now you have an infinite amount of choices. This is the
> same as "... somebody told me that..."
But once I've chosen to tell you truthfully that there's at least
one head, your state of knowledge is the same as it is after my
reply to case 1 - Different route, same destination which, in
the context of the problem, you have already reached.
> Case 3: I tell you: "Please give me some true statement about
> the coins. I'll give you one dollar if you say that at least
> one coin is heads. I'll give you one thousand dollars if you
> say both coins are heads. If you say anything else, or if you
> lie, I'll give you nothing. "
This is plain ridiculous - By introducing the possibity of me
lying, you've embellished the problem into something entirely
different, and irrelevant to this discussion.
Reading Elmo's latest reply, I'm starting to think he may
be using the word "given" in a physical sense, as an actual
instance of throwing a coin, as one might "give it a go",
so you as the observer watch at least the first coin being
thrown, rather than the logical sense of information being
imparted (by whatever means) about two already completed
coin tosses, or somehow confusing the two.
john_r...@sagitta-ps.com
James Dolan wrote:
>
> in article <1128858322.6...@g47g2000cwa.googlegroups.com>,
> <john_r...@sagitta-ps.com> wrote:
>
> James Dolan wrote:
> >
> > ordinary language is forever blurring subtle distinctions
> > like this, but it's even worse to see the mob mindlessly
> > insisting that the distinction doesn't exist than it
> > would be to see someone insisting that ordinary language
> > is unambiguous enough that we should all be able to agree
> > about when the distinction is intended.
>
> |I agree it's wrong to blur subtle distinctions; but where
> |is the distinction between the following statements?
> |
> | "Two coins were flipped and at least one is a head.
> | What are the chances that there are two heads?"
> |
> | "Two coins were flipped. What are the chances that
> | there are two heads, given that at least one is a head?"
> sorry, your question is just too stupid to bother answering.
You needn't bother - I answered it myself in the same breath,
and explained why the difference has no bearing on the problem,
given universally accepted conventions on problem statements:
> |It could be argued that the first version states directly
> |that at least one is a head, whereas in the second this
> |is conveyed by a meta-statement that this fact has been
> |reported or somehow indicated.
> |
> |But within the confines of the questions, we must assume
> |that the relevant information and conditions are equally
> |accurate. So for all practical purposes they differ only
> |in syntax.
and since the source of the "given" meta-statement can be
assumed by convention to be the person posing the problem,
the logical distinction becomes even more vanishingly
insignificant than it already is!
> if you'd really like an answer then i suggest that you read
> what i wrote once more and this time don't hallucinate that
> i said things completely other than what i actually said.
OK, I'll read it again; but I'm pretty sure I understood
your meaning the first time (although it's hard to tell
once those magic mushrooms take effect).
David C. Ullrich
wrote:
>David C. Ullrich <ull...@math.okstate.edu> wrote:
>
>> >I'm 69 years old, and skinny. I'm not a physical threat. I was in his
>> >territory, surrounded by his people. I did not haraass him.
>>
>> Nobody said anything about a physical threat.
>
>Surely you're not implying he was an intellectual threat?
Uh, no, why would you imagine I was implying that?
What I meant by that sentence was exactly what it says:
nobody said anything about a physical threat.
That doesn't even imply that I didn't feel physically
threatened. (And _that_ doesn't imply that I did
feel physically threatened, it's just an example
of one of the many things that are not implied
by that sentence.)
>Victor.
************************
David C. Ullrich
ma...@mimosa.csv.warwick.ac.uk
I don't agree that your state of knowledge is the same after getting the
same replies in Cases 1 and 2. In Case 1 there is no ambiguity at all.
If the I reply "at least one coin is heads" then the proabbility of two
heads is 1/3.
But in Case 2, you have no idea what policy I adopted in choosing
which statement to make. Different policies can lead to different
probabilities. The policy that is often postulated in connection with
this problem is the following. If I see two heads, I will say "at least
one coin is heads". If I see two tails, I will say "at least one coin is
tails". If I see one of each, then I will choose one of the two
statements "at least one coin is heads" and "at least one coin is tails"
at random. If I adopt that policy, and you know I have adopted that
policy, then you can correctly infer that the probability of two heads is
1/2. This policy, which leads to an answer of 1/2 is a perfectly reasonable
one, and so the argument that the answer is 1/2 is no more or less valid
than the argument that it is 1/3. In fact you cannot make any meaningful
estimate of the probability at all.
Derek Holt.
Elmo
1)Two coins were flipped and at least one is x.
2)Two coins were flipped and at least one is y.
1) and 2) are both true statements. x can be heads or tails, y can be
heads or tails, and they are still true statements.
The probability for two like 1) is 1/2.
The probability for two like 2) is 1/2.
3)The probability for two heads, given at least one head is 1/3.
4)The probability for two tails, given at least one tail is 1/3.
In 1) and 2), "at least one is" gives us 1/2. In 3) and 4), "given at
least one" gives us 1/3.
My supposition is that "given at least one" and told" at least one" are
different.
I rest my case.
Eldon
PS I apologize for greeting you in Oklahoma. I should not have done
that. I'm sorry, I wish I had not done that. It's down and done, I
can't change it.
Christian Bau
john_r...@sagitta-ps.com wrote:
> Christian Bau wrote:
> >
> > Case 2: I tell you: "Please give me some true statement about the
> > coins". Now you have an infinite amount of choices. This is the
> > same as "... somebody told me that..."
>
> But once I've chosen to tell you truthfully that there's at least
> one head, your state of knowledge is the same as it is after my
> reply to case 1 - Different route, same destination which, in
> the context of the problem, you have already reached.
Not at all. Don't ever try to play bridge.
> > Case 3: I tell you: "Please give me some true statement about
> > the coins. I'll give you one dollar if you say that at least
> > one coin is heads. I'll give you one thousand dollars if you
> > say both coins are heads. If you say anything else, or if you
> > lie, I'll give you nothing. "
>
> This is plain ridiculous - By introducing the possibity of me
> lying, you've embellished the problem into something entirely
> different, and irrelevant to this discussion.
"If you lie, you get nothing" is an obvious measure so that I don't have
to pay you thousand dollars if both coins are tails and you lie and
claim they are both heads.
Christian Bau
ma...@mimosa.csv.warwick.ac.uk () wrote:
You could even have the following strategy: If there is one head, you
say "at least one coin is heads". If there are two heads, you say "at
most one coin is tails". You never deviate from that rule. You always
use exactly those words.
Even though both statements are logically equivalent, after a dozen
rounds of this game I will be able to guess correctly whether there are
two heads or one head.
Elmo
P[A|B] = P[AB]/P[B].
In this formula, P(B) is defined as the event which has definitely
happened.
When we get 3/4 in the numerator, the answer is 1/3. This defines an
event which has definitely happened, or will definitely happened. It
precisely defines what it means to say "given at least one." It also
precisely defines a coin flip sequence. There is a precise sequence of
events which must happen to get this result.
Alter the numbers in the denominator and get a different answer. When
the denominator equals 1/2, the answer to our question is 1/2. This
precisely defines a different coin flip sequence.
A coin was flipped, it landed heads. I would not assume that heads was
chosen, then the coin was flipped.
Two coins were flipped and at least one landed heads. I would not
assume that heads were chosen, then the coins were flipped. When two
coins were flipped and they landed tails, tails, "at least one is a
tails" is a true statement. When two coins were flipped and they landed
heads, heads, "at least one is a head" is a true statement.
Suppose that we decide to flip for a heads, ie, we will say at least
one heads at hh, ht, and th.
Then, suppose that the coins land tt, and we say, "two coins were
flipped and at least one is a tail." That is a true statement. My
supposition is that "given at least one tail" means something different
from that.
Eldon
Elmo
"I still maintain that to anyone with an adequate grasp
of English and familiar with expressing a condition by
means of a "given" clause (whose use isn't confined to
maths BTW), the two sentences mean exactly the same. "
Elmo starts here:
When two coins are flipped, there are four equalikely outcomes. hh,
ht, th, tt. Any one of which could occur first.
When hh landed first, the "at least one is a head" statement is true.
When tt landed first the "at least on is a tail" statement is true.
Two coins were flipped, given at least one tail, or given at least one
head means something different from that.
Eldon
john_r...@sagitta-ps.com
I understand what you're saying, and it's obviously right
if one takes policies and hidden motives into account.
But, like Christian B with his incentives and lies, in
just mentioning the word "policy" you're extending the
problem beyond its original statement, reading between
lines to introduce aspects which (by convention, we can
assume) aren't there unless explicitly mentioned.
If I asked someone "what is 2 + 2"? they would naturally
reply "4". If I then said "Wrong, nitwit, my 'policy' is
to only ask questions whose answer is to be calculated
in base 3; so the answer is 1", they would be justified
in thinking _I_ was the nitwit for not including this
proviso in the statement of the question!
In fact thinking about it some more, it seems to me
that "X given Y" can quite reasonably be interpreted
as "X assuming Y", even if in principle there is this
meta-statement aspect to the first form.
Elmo
Newsgroups: sci.math
From: David C. Ullrich <ullr...@math.okstate.edu> - Find messages by
this author
Date: Sun, 09 Oct 2005 09:33:39 -0500
Local: Sun, Oct 9 2005 9:33 am
Subject: Re: Me and David C. Ullrich
Reply | Reply to Author | Forward | Print | Individual Message | Show
original | Report Abuse
On Sun, 9 Oct 2005 10:59:55 +0000 (UTC),
Ullrich said>
Exactly what is the difference in meaning between
"Two coins were flipped and at least one is a head. What
are the chances that there are two heads?"
and
"The probability for two heads, given at least one head?"
Elmo started here:
Two coins were flipped and at least one is a head: That is a statement
made about a two coin toss. The coins were tossed and the statement was
made. A statement about one, two coin toss.
"The probability for two heads, given at least one head" is a
mathematical definition. It is precisely defined by a mathematical
formula. It is made about a toss, or a series of tosses. It is defined
by P(B) equals 3/4.
We can make the given statement prior to the toss. Suppose that we make
the "given at least one head" statement, then toss and get tt. We can
say "two coins were tossed and at least one is a tail". That's a true
statement. It does not mean the same thing as "given at least one
tail."
Eldon
Ross A. Finlayson
Boo!
Ha ha ha ha.
Ullrich: hedgehog. That's affectionate, damnit. Ullrich, is like a
Volkswagen bus. Jetta: water cooled.
Hey, if you get bored, know what you're talking about, and want to talk
about mathematics and infinity on the phone you can telephone me:
208/476-3831.
I'm still waiting for the management summary of differential equations.
Han says there'll never be one. How about a list of all the named
non-linear equations? Now Winter, Winter is good at billiards. Bau is
quite competent, but that's not an innocuous non-insult.
Seems like you have a shell game there, Elmo. What are you going to
do, argue against Marilyn Mach vos Savant, the woman with the highest
IQ who writes in the weekly color Parade magazine newspaper insert with
the advertisements for the collectable (collectible) plates? Others
do.
So, be careful Elmo, or you might find yourself arguing against one of
the most widely read authors in the plebeian bourgeousie
plate-collecting soccer mom light SUV never cancelled their
subscription to Reader's Digest semi-literary sphere.
Ross
--
"The alternative, of a finite
number of scales and a lowest "layer of the onion" would,
when considered rightly, amount to no more than a giant
static diagram from Euclid, maybe with 2^10000 lines or
a number too large to print in the most compact notation
known to man if all the oceans were ink, but finite all
the same, and thus utterly preposterous!" - Ramsden
Golf course ate my balls.
john_r...@sagitta-ps.com
Elmo wrote:
>
> Dr Ramsden said
> "I still maintain that to anyone with an adequate grasp
> of English and familiar with expressing a condition by
> means of a "given" clause (whose use isn't confined to
>
> Elmo starts here:
> When two coins are flipped, there are four equalikely
> outcomes. hh, ht, th, tt. Any one of which could occur first.
>
> When hh landed first, the "at least one is a head" statement is true.
> When tt landed first the "at least on is a tail" statement is true.
>
> Two coins were flipped, given at least one tail, or given at least one
> head means something different from that.
OK, that confirms my suspicion that you think, possibly
subconsciously, of these two forms of words as indicating
some difference in the timing of the probability decision.
One form of expression, I think you assume, implies a kind
of pending probability problem in which a decision is made
on the second coin after only the first has been flipped.
Obviously in that case the probability of the second
coin landing heads is 1/2, whatever you are told about
the first, and in fact any such information is by then
irrelevant.
But everyone else (I think it's safe to say, as far as
_anything_ is safe to say about this damned problem,
which is rapidly becoming as confusing as the goats
behind the doors problem!) takes it for granted that
the coins have both already been flipped when you're
asked to state the probability.
In short, with either form of words the problem asks
for the "post hoc" probability, once both coins have
been flipped, with/given/assuming extra information
that at least one coin _has_ landed heads.
Elmo
|distinction between the following statements?
|
| "Two coins were flipped and at least one is a head. What are the
| chances that there are two heads?"
|
| "Two coins were flipped. What are the chances that there are two
| heads, given that at least one is a head?"
|
flipped, and at least one of the coins has been inspected. There is
definitely one head. There is no evidence that the writer of the
question knows anything of the second coin.
The second statement says that the two coins have been tossed. "Given
that at least one is a head" means that "heads" have been chosen, and
that all tails, tails flips will be rejected. For tt to be rejected,
both coins will have to be inspected.
That's different, it ain't too subtle.
Eldon:)
john_r...@sagitta-ps.com
Christian Bau wrote:
>
> In article <1128873353.7...@o13g2000cwo.googlegroups.com>,
> john_r...@sagitta-ps.com wrote:
> >
> > Christian Bau wrote:
> > >
> > > Case 2: I tell you: "Please give me some true statement about the
> > > coins". Now you have an infinite amount of choices. This is the
> > > same as "... somebody told me that..."
> >
> > But once I've chosen to tell you truthfully that there's at least
> > one head, your state of knowledge is the same as it is after my
> > reply to case 1 - Different route, same destination which, in
> > the context of the problem, you have already reached.
>
> Not at all. Don't ever try to play bridge.
Again, you're letting real life considerations intrude on what
is an abstract problem, seeing hidden depths that aren't there
(unless explicitly stated, in which case it becomes a different
problem).
Joubin Houshyar
> 2)Two coins were flipped and at least one is y.
>
> 1) and 2) are both true statements. x can be heads or tails, y can be
> heads or tails, and they are still true statements.
(Your use of 'x' and 'y' is counter productive here and is causing your
confusion. T or H would do fine.)
So:
> 1)Two coins were flipped and at least one is Tails.
> 2)Two coins were flipped and at least one is Heads.
How do you know they are "true statements"?
Based on ?what? 'bit of information' do you accept that "at least one
is [Heads]"?
How is that true?
Why is the possibility that "two coins were flipped and both are Tails"
is equal to nil in statement(1)? (It has to be nil for statement(1) to
be "true"...)
Conversely, why would you accept as "true" the statement(2) that "Two
coins were flipped and at least one is Heads"? (Why? Is the
possibility of "Two Tails" somehow rendered nil?)
> My supposition is that "given at least one" and told" at least one" are
> different.
That really makes no sense, Eldon.
If someone says "I have thrown two coins and at least one is a head",
you should (shouldn't you?) quickly ask: How do you know "at least one
is a head"?
Then, that person will reply: Oh, that is a "given", since I always
re-throw if they are both tails.
So "at least one is" -> "given one is".
Unless you are prepared to accept that it is impossible for both to be
Tails, based on ... ... based on what exactly?
2 possibilities:
1) They are 'Magic' coins and never land Tail/Tail. That would require
a new branch of probability dedicated to Magical Processes
2) There is a 'Process' that halts only when "at least one" of the
coins is Heads. In which case you are back on terra firma and the
probability of the toss being Heads/Heads is 1/3.
>
> I rest my case.
Think, Eldon. Think clearly.
>
> PS I apologize for greeting you in Oklahoma. I should not have done
> that. I'm sorry, I wish I had not done that. It's down and done, I
> can't change it.
I think his objection was not to your "greeting" but to your 'stalking'
him.
Christian Bau
john_r...@sagitta-ps.com wrote:
> I understand what you're saying, and it's obviously right
> if one takes policies and hidden motives into account.
> But, like Christian B with his incentives and lies, in
> just mentioning the word "policy" you're extending the
> problem beyond its original statement, reading between
> lines to introduce aspects which (by convention, we can
> assume) aren't there unless explicitly mentioned.
Well, I mentioned them explicitely. "Lies" were mentioned only insofar
as they were _explicitely_ excluded.
john_r...@sagitta-ps.com
God made men - and Sam Colt made them equal!
Not suggesting for a moment you would go "postal",
in fact you sound like a decent enough guy; but to
put it bluntly, how was Ullrich to know you weren't
a nutter who might? It has been known ..
john_r...@sagitta-ps.com
john_r...@sagitta-ps.com wrote:
>
> Elmo wrote:
> >
> > Dr Ramsden said
> > "I still maintain that to anyone with an adequate grasp
> > of English and familiar with expressing a condition by
> > means of a "given" clause (whose use isn't confined to
> > maths BTW), the two sentences mean essentially the same."
> >
> > Elmo starts here:
> > When two coins are flipped, there are four equalikely
> > outcomes. hh, ht, th, tt. Any one of which could occur first.
> >
> > When hh landed first, the "at least one is a head" statement is true.
> > When tt landed first the "at least on is a tail" statement is true.
> >
> > Two coins were flipped, given at least one tail, or given at least one
> > head means something different from that.
>
> OK, that confirms my suspicion that you think, possibly
> subconsciously, of these two forms of words as indicating
> some difference in the timing of the probability decision.
Got it now - Your post that started this thread begins:
> On this forum I have argued the question, "Two coins were
> flipped and at least one is a head. What are the chances
> that there are two heads?"
I put it to you that this, in your mind, is equivalent to:
> On this forum I have argued the question, "Two coins were
> flipped and after one is a head, what are the chances
> that the second will be a head?"
Elmo
Lets say two coins have been thrown, I can't see the result, but you
can. I want to find out a bit more. I ask a question and you have to
answer thruthfully.
Case 1: I ask "Is at least one coin heads?",
Eldon starts here:
Examine what you've done. You have selected Heads, prior to knowledge
of the outcome. This preselection is the vital ingrediant to flip two
coins three equally likely ways.
In our formula, P(hh|at least one head) = 1/3, we say that P(at least
one head) equals 1/4 + 1/4 + 1/4 + 0 equals 3/4.
In other words P(at least one head|hh) = P(at least one head|ht).
For the latter to be true, there must be a preselection of heads.
When there was no preselection, when heads were chosen after the
inspection
Then:
P(told"at least one head"|hh) = 2P(told"at least one head"|ht)
Suppose that you ask the question and the answer is "no". Then we know
that "Two coins were tossed and at least one is a tail. The chances for
two tails is one". (That is different from "the probability for two
tails, given at least one tail".
Eldon:)
Elmo
> Ha ha ha ha.
>
>
> Seems like you have a shell game there, Elmo. What are you going to
> do, argue against Marilyn Mach vos Savant, the woman with the highest
> IQ who writes in the weekly color Parade magazine newspaper insert with
> the advertisements for the collectable (collectible) plates? Others
> do.
>
> So, be careful Elmo, or you might find yourself arguing against one of
> the most widely read authors in the plebeian bourgeousie
> plate-collecting soccer mom light SUV never cancelled their
> subscription to Reader's Digest semi-literary sphere.
>
I have argued with Marilyn already. She says that the key to deciding
whether the answer is 1/3, or 1/2, is whether or not there is an "at
least one is" statement. That's wrong, and I've proved it. She has
refused to acknowledge my argument.
The key to flipping two coins three ways is to decide upon a null,
prior to the inspection.
Example:
Two coins were flipped until at least one is a head. The probability
for two heads is 1/3.(the until tells us that there was preselection,
the writer will reflip at tt. In order to do so the writer must inspect
both coins)
Two coins were flipped, at least one is a tail. The probability for two
tails is 1/2. (no preselection, no evidence that the writer of the
statement has inspected more than one coin)
Eldon:)
> Ross
>
Jon Slaughter
"Elmo" <elmo...@yahoo.com> wrote in message
news:1128780755.1...@g44g2000cwa.googlegroups.com...
> On this forum I have argued the question, "Two coins were flipped and
> at least one is a head. What are the chances that there are two heads?"
>
> heads, given at least one head?"
>
um...
does "the chance that there are two heads"
mean the same as " The probability for two heads"?
if so let x represent that idea.
is "given atleast one head" the same as "atleast one is a head"?
if so let y reprsent that idea
hence your statements are
Two coins were flipped and y => x
[Two coins were flipped] x <= y
where [..] means implied(which is obvious in this case)
hence your statements are equivilent assuming we agree on the equivilence of
x and y.
you pointed out, I think, in another post that that in all actuality the
statements that I called y are actually different.
i.e.
"given atleast one head" != "atleast one is a head"?
but how can this be? You must explain it to me in terms of the english
language and not in terms of probability as it has nothing to do with
probability.
"given that I have a glass of milk in my hand" != "I have a glass of milk in
my hand"???
are you some how implying that the two are different because one is less
likely to be true? Ofcourse one says that I want you to ASSUME that what I
have said is true and the other is TELLING you what I have said is true. In
either cause though you have been told the statement "I have a glass of milk
in my hand" as being true. In either case I could be lieing. Just because
one wants you to assume that and the other "demands" it has nothing to do
with the issue.
It seems to me that you are interpreting the two statements as being
temporally different. One is refering to a future event while the other to
a past/present one.
How would you interpret the logical syllogism
Given all humans are not people
all people are humans
and
all apples taste delicious
all delicious things are healthy
?
Do you think the logic behind these two are distinctly different?
Jon
Elmo
> > 1)Two coins were flipped and at least one is x.
> > 2)Two coins were flipped and at least one is y.
> >
> > 1) and 2) are both true statements. x can be heads or tails, y can be
> > heads or tails, and they are still true statements.
>
> (Your use of 'x' and 'y' is counter productive here and is causing your
> confusion. T or H would do fine.)
>
statement. We can always make an "at least one is y" statement. We
cannot always make an "at least one is T" or "at least one is H".
> So:
>
> > 1)Two coins were flipped and at least one is Tails.
> > 2)Two coins were flipped and at least one is Heads.
>
> How do you know they are "true statements"?
>
> Based on ?what? 'bit of information' do you accept that "at least one
> is [Heads]"?
>
We always assume a true statement. When we assume a non true statement,
we are assuming a different question. One event happened, another event
was reported.
> How is that true?
>
Just like the statement said, "Two coins were flipped, and at least one
is a head." We know that the statement was made, and that the statement
is true. We know that hh happened, and the statement was made, or, ht
happened and the statement was made, or th happened and the statement
was made. (notice the or's)(we also know that the statement was made
after some kind of inspection. The writer of the statement had to have
inspected at least one coin)
> Why is the possibility that "two coins were flipped and both are Tails"
> is equal to nil in statement(1)? (It has to be nil for statement(1) to
> be "true"...)
>
When the tails statement was made, we know that hh didn't happen.
> Conversely, why would you accept as "true" the statement(2) that "Two
> coins were flipped and at least one is Heads"? (Why? Is the
> possibility of "Two Tails" somehow rendered nil?)
>
When the heads statement was made, we know that tt didn't happen.
>
> > My supposition is that "given at least one" and told" at least one" are
> > different.
>
> That really makes no sense, Eldon.
>
It makes sense, only if you understand what I'm telling you. An
elementary rule of debating is to understand the other guy's argument.
If you wish to negate my argument, first you should understand it. I
used 'x' and 'y' for a reason. Understand it.
> If someone says "I have thrown two coins and at least one is a head",
> you should (shouldn't you?) quickly ask: How do you know "at least one
> is a head"?
>
Suppose that two coins were thrown and they landed tt. The statement
then was "Two coins were thrown and at least one is a tail." Bruce the
bettor, only heard the statement. He bet for two heads and will win.
What odds should he collect?
It's silly to assume that I should quickly ask anything. I should
answer the question as written. When the reflipper is going to reflip
on a predetermined outcome, this is the necessary preselection of which
I speak. The necessary preselection to flip two coins three equally
likely ways.
> Then, that person will reply: Oh, that is a "given", since I always
> re-throw if they are both tails.
>
This is added information which was not, is not part of our question.
> So "at least one is" -> "given one is".
>
> Unless you are prepared to accept that it is impossible for both to be
> Tails, based on ... ... based on what exactly?
>
When two coins are thrown there are four equalikely outcomes. FOUR.
> 2 possibilities:
>
> 1) They are 'Magic' coins and never land Tail/Tail. That would require
> a new branch of probability dedicated to Magical Processes
>
Methinks you've confused yourself and are making my argument. When two
coins are thrown, tt is one of the four equally likely.
> 2) There is a 'Process' that halts only when "at least one" of the
> coins is Heads. In which case you are back on terra firma and the
> probability of the toss being Heads/Heads is 1/3.
>
Preselect heads, then flip.
Example:
Two coins were flipped until at least one is a head. The probability
for two heads is 1/3. (The 'until' is necessary)
Eldon
David Ames
Elmo wrote:
> Two coins were flipped and they landed xy.
>
> 1)Two coins were flipped and at least one is x.
> 2)Two coins were flipped and at least one is y.
>
Your terms of discussion have changed. Originally you allowed Heads
and Not-Heads, and wanted to know about the likelihood of two events
both being Heads. That sample space consisted of Heads, Tails, and
On-Edge. Now you only allow two out of three possibilities.
David Ames
Elmo
>
> "given atleast one head" != "atleast one is a head"?
>
> but how can this be? You must explain it to me in terms of the english
> language and not in terms of probability as it has nothing to do with
> probability.
>
> "given that I have a glass of milk in my hand" != "I have a glass of milk in
> my hand"???
>
Put two coins on a table. One with the heads up, and one with the
tails.
1. There are two coins, at least one is a head.
2. There are two coins, at least one is a tail.
Those are both true statements. With the information we have at hand,
they are equalikely statements. (there is no known reason to say one,
over the other)
When statement 1 is said, the hearer of the statement knows nothing of
a tails.
When statement 2 is said, the hearer knows nothing of heads.
To say, "What are the chances for two heads? would not be an
appropriate question, as the coins were not tossed, we placed them
there. Never the less, the "at least one is statements are true.
The above demonstrate pretty well what it means to say "at least one
is".
There is a probability formula which states that, "When two coins are
flipped, the probability for two heads, given at least one head is
1/3." It's an abstract mathematical formula. We could change the
numbers in the formula, and alter what it means to say "given at least
one." This would not alter what it means to say "at least one is."
I know of no mathematical formula to define "given a glass of milk in
my hand."
Given at least one heads means that we'll have a success every time at
hh, ht, and th. At least one is a head means whatever it says. It isn't
affected by that little formula, that little mathematical definition.
At ht and th, both statements are true. To assume that with the heads
statement the probability for two heads is 1/3, and with the tails
statement the probability for two tails is 1/3 is erroneous. It can't
happen.
Eldon
Shmuel (Seymour J.) Metz
at 10:59 AM, jdo...@math-cl-n03.math.ucr.edu (James Dolan) said:
>"what's the probability of x given y?" and "what's the probability of
>x given that someone tells you that y?" can (of course)
There is no "of course"; you're begging the question.
>have quite different answers (and other similar phrasings can have
>further different answers). ordinary language is forever blurring
>subtle distinctions like this, but it's even worse to see the mob
>mindlessly insisting that the distinction doesn't exist
If the questioner intended for the questions to have different
meanings then it was his responsibility to ask the questions in a
fashion precise enough for the respondent to understand what meaning
he intended for each question. Of course, if you're thinking about the
possibility that someone is lying, then the question is ill formed
without a probability distribution for someone's truthfulness.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
Elmo
mensa...@aol.compost wrote:
> Elmo wrote:
> > On this forum I have argued the question, "Two coins were flipped and
> > at least one is a head. What are the chances that there are two heads?"
> >
> > heads, given at least one head?"
> >
> > I say that "given at least one head", and told, "at least one head"
> > mean two different things.
>
> Really?
>
> >
> > Dr. Ullrich has stated in this forum, that the two mean the same thing.
> > If for no other reason than that he said so. Also because everyone
> > assumes them to be the same. (everyone does not, I don't)
>
> Well, mathematics is not a democracy. But it isn't a
> parlimentary system either. There are no minority opinions
> allowed, only Truth. And if you think your opinion is Truth,
> then you have two choices:
>
> - prove you are right
>
> or
>
> - prove the majority is wrong
>
> >
>
> >
> > The probability for two tails, given at least one tail is one third.
> > That is defined mathematically and is not arguable.
>
> Then why are you arguing?
>
I am not arguing with the mathematical definition. It's true by
mathematical definition. If it's desired for it to mean the same thing
as "told at least one", then they should change the definition.
> >
> > What it means to say, "at least one is a tail" is defined in the world
> > domain and means what it says. As the math domain is a subset of the
> > world domain, it is not possible to re-define, inside the math domain,
> > what "at least one is" means in the outside world.
>
> Huh?
>
By your reply "Huh?", I assume that you did not understand what I said.
What I said is the truth, if you wish to disprove it, first you should
understand it.
Math domain is a subset of the world domain. (did you not understand
that?)
A definition inside the math domain is not necessarily valid outside
the math domain. (is that not true, or not understandable?)
A statement made outside the math domain has to mean what it says
outside the math domain.
Example:
In a family newspaper, the question:
Two coins were flipped and they landed both tails. The statement was
generated, "Two coins were flipped and at least one is s tail. What are
the chances for two tails?" Bruce only heard the statement, and bet for
two tails. He will win, what odds should he collect?
Should our little definition affect what their statement "at least one
is" meant?
If you wish those two to mean the same, change the definition.
> >
> > An example:
> >
> > Suppose that a computer program randomizes the coins, or dice, or puppy
> > dogs. Then shows us to colors. Suppose that we see a red and an orange.
> >
> > We can make the statement"there are two colors and at least one is red"
> > or we can make the statement "there are two colors and at least one is
> > orange." The red statement tells nothing of orange, and the orange
> > statement tells nothing of red.
>
> Ok.
>
> >
> > The statements "at least one is red" and "at least one is orange"
>
> Wait a minute. That's TWO statements.
>
> > demonstrate what it means to say "at least one is".
>
That is TWO statements. They are both true. They each demonstrate what
it means to say "at least one is". That's why I have the 'and' between
them.
Eldon
Jon Slaughter
Who said there was? Also, just because you don't "know" doesn't mean that it
doesn't exist.
>
> Given at least one heads means that we'll have a success every time at
> hh, ht, and th. At least one is a head means whatever it says. It isn't
> affected by that little formula, that little mathematical definition.
>
This makes no sense. Try stating your questions in a more mathematical way.
> At ht and th, both statements are true. To assume that with the heads
> statement the probability for two heads is 1/3, and with the tails
> statement the probability for two tails is 1/3 is erroneous. It can't
> happen.
>
> Eldon
>
Hmm, I thought my examples and questions were pretty simple and straight
forward but you failed to answer them...
Just curious... Has it ever crossed your mind that maybe the problem has
nothing to do with math but with semantics and possibly the ambiguity of the
english language? Remember that we use natural languages to describe the
mathematical language. Surely there will be some potential for confusion?
http://www.winlab.rutgers.edu/~crose/545_html/stochastic1/node2.html
(you should read the following link carefully as I think it addresses the
heart of the matter)
http://en.wikipedia.org/wiki/Bayesian_inference
http://en.wikibooks.org/wiki/Probability:Probability_Spaces
http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html
Try to state the problem exactly in terms of the notation above. I.e.,
define your probability space and show the exactly mathematical computations
on how you arrive at the following:
P(hh|at least one h) = 1/3 to
P(hh|told "at least one h) = 1/2.
To me, the statements on the LHS are EXACTLY THE SAME. Just because you
added "told" does not change anything and has no mathematical basis. "at
least one" and "told atleast one" HAVE THE EXACT SAME ENGLISH
MEANINGS!!?!?!?! Prove otherwise if you don't agree! Realize that if you
have some problem with what I said then you are not dealing with math issues
but english.
Looking on the rhs I can only guess as to how you got the numbers.
If I let my sample space O = {hh,ht,th,tt}
and
P(hh) = 1/4
P(ht) = 1/4
P(th) = 1/4
P(tt) = 1/4
P(s = hh | s = ht, s = th or s = hh) = 1/3
Its obvious why its 1/3 but lets check
P(s = hh | s = ht, s = th or s = hh) = P(s = ht, s = th or s = hh | s =
hh)*P(s = hh)/P(s = ht, s = th or s = hh) = 1*1/4/(3/4) = 1/3
how to get 1/2???
well, lets do this experiment:
What is the probability that flipping a quarter first then a penny second
will result in the quarter and penny landing on heads given that the quarter
***landed*** on heads?
O = {qh ph, qh pt, qt ph qt pt)
so we are looking for
P(penny = heads | quarter = heads) ??
we see that there are 2 possible choices where the quarter = heads and only
one of them has the penny as heads also.. hence the probability is 1/2
now turn the penny into a quarter, which doesn't ofcourse effect the
probability.
Your issue then is obivously one of distinguishability. If you distingish
the two quarters then it translates into "probability that FIRST quarter
flipped is a head" as compared to the indistingishable case "probability
that either the FIRST OR SECOND flipped is a head"!!!! Thats a huge
difference.
i.e., your problem as you have stated isn't "atleast one is head" but
"atleast one is heads" and "atleast the first is heads" but since you leave
out the "first". How you are doing this, it seems, is that you are changing
the "rules" after the first flip i.e., switching it from unordered to
ordered(or vice versa) after the first flip to arrive at a different answer.
again, you seem to be saying that
"given atleast one" means that "the first one"
and
"atleast one" means "either first or second"
you think that because we say "given" we must have flipped the first already
and hence we "ignore" its probability because it already happend and there
is no reason to take it into account... a temporal problem.
your problem is perfectly valid to some extend but erroneous in the sense
that you are taking into account something that you shouldn't. You are
misinterpreting the semantics of the sentence.
i.e.
hopefully this gets directly at the problem.
I flip a quarter and it lands and I see the the actual side of the quarter
but you have not seen it. I ask you "What is the probability that the
quarter is on heads" and you answer 1/2... I SAY "NO YOU IDIOT, ITS 0
BECAUSE IT LANDED ON TAILS!!!!"... and you look at me like I'm retarded
because you know that there is a 1 out of 2 chance it would have landed on
tails.
i.e.
P(heads | heads) != P(heads)
(your problem is just a step away from this by added another variable but
the issue is basicaly the same so it looks more rational than the above).
As they say, You are comparing apples and oranges(posterior and priori
probability)
Anyways, thats my guess.
Jon
Elmo
> Elmo wrote:
> >
> > Dr Ramsden said
> > "I still maintain that to anyone with an adequate grasp
> > of English and familiar with expressing a condition by
> > means of a "given" clause (whose use isn't confined to
> > maths BTW), the two sentences mean essentially the same."
> >
> > Elmo starts here:
> > When two coins are flipped, there are four equalikely
> > outcomes. hh, ht, th, tt. Any one of which could occur first.
> >
> > When hh landed first, the "at least one is a head" statement is true.
> > When tt landed first the "at least on is a tail" statement is true.
> >
> > Two coins were flipped, given at least one tail, or given at least one
> > head means something different from that.
>
> OK, that confirms my suspicion that you think, possibly
> subconsciously, of these two forms of words as indicating
> some difference in the timing of the probability decision.
>
> One form of expression, I think you assume, implies a kind
> of pending probability problem in which a decision is made
> on the second coin after only the first has been flipped.
>
> Obviously in that case the probability of the second
> coin landing heads is 1/2, whatever you are told about
> the first, and in fact any such information is by then
> irrelevant.
>
are the chances for two heads?
We have a statement. The statement constitutes the entire question.
Two coins were tossed. How do we know? The statement told us so.
At least one is a head. What does that mean? It means tt didn't happen.
Because of the statement, we know that hh happened and the statement
was made, Or,
ht happened and the statement was made, Or,
th happened and the statement was made.
Notice the Or's, we can take them individually and we have:
Two coins were tossed and they landed hh, the statement was made, "Two
coins were tossed and at least one landed heads. What are the chances
for two heads?" Bruce only heard the statement, he bet for hh. He will
win, what odds should he collect. The odds should depend upon the
correct answer to the question. If the answer is 1/3, he should get two
to one.
OR:
We have:
Two coins were tossed and they landed ht, the statement was made, "Two
coins were tossed and at least one is a head. What are the chances for
two heads?" Bruce only heard the statement, he bet for hh, he will
lose. The statement, "Two coins were tossed and at least one is a tail.
What are the chances for two tails?" would have also been true. As both
statements are true here, would they each have the same answer? Bruce
will lose either way, he doesn't care.
OR: They landed th and we have a scenario just like ht. As either the
"at least one is a head" and "at least one is a tail" statement would
be true, would they be equally likely to be said, or would one take
precedent over the other?
When the coins were tossed and they landed one of each. The "at least
one is a head" and the "at least one is a tail" statement would be
true. Would the "given at least one head" and the "given at least one
tail" statement both be true?
When the coins were tossed and they landed hh, the "at least one is a
head" statement is true, would the "given at least one head" also be
true?
Suppose that the coins had landed tt. The "at least one is a tail"
statement would have been true. Would the "given at least one tail"
have also been true, and meant the same thing?
> But everyone else (I think it's safe to say, as far as
> _anything_ is safe to say about this damned problem,
> which is rapidly becoming as confusing as the goats
> behind the doors problem!) takes it for granted that
> the coins have both already been flipped when you're
> asked to state the probability.
>
We can say, "Two coins will be flipped. Given at least one head, the
probability for two heads is 1/3. Given at least one tail, the
probability for two tails is 1/3." We can say that "Two coins will be
flipped, the probability for at least one head is 3/4. The probability
for at least one tail is 3/4." We cannot say, "Two coins will be
flipped, 'at least one is a tail' will be true."
Therefore, we know that our statement, made as a true statement, was
made after the toss.
Hope that clears it up. It isn't that confusing, actually. The goats
behind the doors isn't that tough either. I can explain it also. Just
remember that words mean things.
This email demonstrates the difference between the two statements in
question. They definitely have different meanings, if the words mean
what the words say. If they're the same because Doctor Ullrich says
they're the same, then, that's a different argument.
Eldon
Dik T. Winter
...
> Given at least one heads means that we'll have a success every time at
> hh, ht, and th. At least one is a head means whatever it says.
No, that is not what is implied. It is implied that the current toss
has at least one head. Nothing more, nothing less. But you are talking
about semantics rather than mathematics. You assume that, in some way,
the sentence "given one is head" means that it would also be possible
that the sentence would have been "given one is tail". Or in one of it's
many disguises.
Good. Let me formulate it differently. Two coins are tossed, and it
is announced that one is either heads or tails. What is the probability
that the other is the same? The probability is 1/2.
Another formulation. Two coins are tossed and it is announced that one
is heads, or nothing is announced. What is the probability that both are
equal? Assuming that there is an announcement when at least one head
crops up, the probability is again, 1/2.
A final formulation. Two coins are tossed and it is announced that one
is heads, or nothing is announced. Assuming that there is an announcement
when at least one head crops up, what is the probability that, given such
an announcement, both are the same? 1/3.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
mensa...@aol.com
Why should the definition be changed? Just to suit you?
It's simpler and wiser to have one defintion and have everyone's
thinking conform to it rather than try to make a definition that
conforms to everyone's thinking.
>
> > >
> > > What it means to say, "at least one is a tail" is defined in the world
> > > domain and means what it says. As the math domain is a subset of the
> > > world domain, it is not possible to re-define, inside the math domain,
> > > what "at least one is" means in the outside world.
> >
> > Huh?
> >
> By your reply "Huh?", I assume that you did not understand what I said.
Or it could mean I can no more think illogically than I can
think in French.
> What I said is the truth, if you wish to disprove it, first you should
> understand it.
>
> Math domain is a subset of the world domain.
Maybe it is, maybe it isn't.
> (did you not understand that?)
Maybe I don't agree with it.
> A definition inside the math domain is not necessarily valid outside
> the math domain. (is that not true,
No, not if the math domain is a subset.
> or not understandable?)
You've got it backwards, the real world is a subset of the math
world. There may be math that has no real world correlation but
there is nothing in the real world outside the domain of math.
>
> A statement made outside the math domain has to mean what it says
> outside the math domain.
> Example:
> In a family newspaper, the question:
> Two coins were flipped and they landed both tails. The statement was
> generated, "Two coins were flipped and at least one is s tail. What are
> the chances for two tails?"
But this is not a question of probability. The probality function
has collapsed and the values are set even if they are not known.
> Bruce only heard the statement, and bet for
> two tails. He will win, what odds should he collect?
But there are no odds, since the probability function has collapsed.
What you CAN say is: what is the probability Bob can pick the
correct outcome having heard the statement. THAT function collapses
when Bob makes his pick. The payoff should be based on the function
that has not collapsed, not on the one that has.
>
> Should our little definition affect what their statement
> "at least one is" meant?
No, you don't understand the definition.
> If you wish those two to mean the same, change the definition.
The definition can't be changed EVERY time someone doesn't
understand it.
>
> > >
> > > An example:
> > >
> > > Suppose that a computer program randomizes the coins, or dice, or puppy
> > > dogs. Then shows us to colors. Suppose that we see a red and an orange.
> > >
> > > We can make the statement"there are two colors and at least one is red"
> > > or we can make the statement "there are two colors and at least one is
> > > orange." The red statement tells nothing of orange, and the orange
> > > statement tells nothing of red.
> >
> > Ok.
> >
> > >
> > > The statements "at least one is red" and "at least one is orange"
> >
> > Wait a minute. That's TWO statements.
> >
> > > demonstrate what it means to say "at least one is".
> >
> That is TWO statements. They are both true. They each demonstrate what
> it means to say "at least one is". That's why I have the 'and' between
> them.
But "at least one is" is true even when BOTH colors are orange.
Therefore, "at least one is" does NOT mean the colors are different.
>
> Eldon
Elmo
>
> >
> > Given at least one heads means that we'll have a success every time at
> > hh, ht, and th. At least one is a head means whatever it says. It isn't
> > affected by that little formula, that little mathematical definition.
> >
>
> This makes no sense. Try stating your questions in a more mathematical way.
>
P(A|B)=P(B|A)P(B) / P(B) Where B is defined as the event which has
happened)
Then P(B) is the probability that said event happened.
P(hh|at least one head) = (1/4)/P(B)
Our sample space for a two coin toss is hh, ht, th, tt. They are
1/4+1/4+1/4+1/4.
When P(B) = 1/4+1/4+1/4+0 = 3/4
(1/4)/(3/4) = 1/3.
In English, The probability for two heads, given at least one head is
1/3.
Note that if we alter P(B) to 1/4+1/2*1/4+1/2*1/4+0=1/2
Then P(hh|B)= 1/2.
We get 1/3, or 1/2, depending upon whether or not we have factors in
the denominator. When we have factors, the probability for hh = 1/2. No
factors, 1/3. Why would we have, or not have factors? The different
P(B)'s define two different coin flip sequences.
When we say "given at least one" we have no factors and we get 1/3. Why
do we have no factors? Why do we get a quarter probability for ht, just
like we do at hh? We say "at least one is a head" every time at hh, and
every time at ht, because that's what it means to say "given at least
one". To say "at least one head" every time at ht, heads must have been
chosen, prior to the toss.
Make the decision after the toss, when the writer of the statement
looks at the toss without preference to heads or tails, then with ht,
or th, tails and heads are equally likely to be chosen. When they are
equally likely to be chosen, then there are factors in the denominator,
and the answer is 1/2.
When the coins were tossed, and the statement was made, then there are
factors. To do a working model for the definition, to remove the
factors, the heads/tails decision must be made prior to the toss.
> > At ht and th, both statements are true. To assume that with the heads
> > statement the probability for two heads is 1/3, and with the tails
> > statement the probability for two tails is 1/3 is erroneous. It can't
> > happen.
> >
> > Eldon
> >
>
> Hmm, I thought my examples and questions were pretty simple and straight
> forward but you failed to answer them...
>
> Just curious... Has it ever crossed your mind that maybe the problem has
> nothing to do with math but with semantics and possibly the ambiguity of the
> english language? Remember that we use natural languages to describe the
> mathematical language. Surely there will be some potential for confusion?
>
This is a little probability question that has been around for years.
It has everything to do with what it means to say "at least one is".
There are a lot of ambiguities in the arguments. The question is
stripped to bare essentials and is not ambiguous. There is a lot of
potential for confusion. Case in point look at all the mathematical
people who look at my arguments, and are either confused, or act
confused. It's a counter intuitive question. Looks like a third, but
answers a half. People caught on the naive side have a tendency to act
confused. Ambiguity is a common plea. "Didn't intend to be answering
that question" or "Don't think the asker of the question intended
that".
>
> http://www.winlab.rutgers.edu/~crose/545_html/stochastic1/node2.html
>
> (you should read the following link carefully as I think it addresses the
> heart of the matter)
> http://en.wikipedia.org/wiki/Bayesian_inference
>
> http://en.wikibooks.org/wiki/Probability:Probability_Spaces
>
> http://www.probability.net/
>
> http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html
>
>
> Try to state the problem exactly in terms of the notation above. I.e.,
> define your probability space and show the exactly mathematical computations
> on how you arrive at the following:
>
> P(hh|at least one h) = 1/3 to
> P(hh|told "at least one h) = 1/2.
>
The difference is whether or not there are factors in the denominator.
The key is whether the statement was made, and then the coins were
flipped, or whether the coins were flipped and then the statement was
made.
When the coins were flipped, then the statement, the answer is 1/2. We
have factors at ht, and th.
To remove the factors, the statement must have been made first,
however, the statement can't be made first, as a true statement.
> To me, the statements on the LHS are EXACTLY THE SAME. Just because you
> added "told" does not change anything and has no mathematical basis. "at
> least one" and "told atleast one" HAVE THE EXACT SAME ENGLISH
> MEANINGS!!?!?!?! Prove otherwise if you don't agree! Realize that if you
> have some problem with what I said then you are not dealing with math issues
> but english.
>
Told at least one and given at least one may have the same English
meanings. Given at least one has a special mathematical meaning. If you
wish them to have the same mathematical meaning, change the
mathematical definition.
> Looking on the rhs I can only guess as to how you got the numbers.
>
> If I let my sample space O = {hh,ht,th,tt}
>
> and
>
> P(hh) = 1/4
> P(ht) = 1/4
> P(th) = 1/4
> P(tt) = 1/4
>
> P(s = hh | s = ht, s = th or s = hh) = 1/3
>
> Its obvious why its 1/3 but lets check
>
> P(s = hh | s = ht, s = th or s = hh) = P(s = ht, s = th or s = hh | s =
> hh)*P(s = hh)/P(s = ht, s = th or s = hh) = 1*1/4/(3/4) = 1/3
>
There are no factors. s is the same at hh, ht, th. The writer looks at
ht, or th and picks h every time. That removes the factors. To pick h
every time at ht and th, h has to have been already chosen. Most who
were caught on the naive side don't wish to go through the process to
understand why this must be so. How do you have to flip the coins to
make a working model for P(hh|at least one)=1/3? What makes the
distinction? At least one is a head can be stated at hh, ht, or th. At
least one is a tail can be stated at tt, th, or ht. We can always have
an "at least one is" statement. We can't always pay two to one. To pay
two to one, the null must have been decided, prior to the toss.
>
> how to get 1/2???
>
Put the factors in the denominator. When the coins were tossed, and
then the heads/tails decision was made, the answer is 1/2. It was 1/2
because there are factors in the denominator. At ht, and th, the
flipper was equally likely to pick heads, or tails. When there are no
factors, the flipper already knew the choice.
> well, lets do this experiment:
>
> What is the probability that flipping a quarter first then a penny second
> will result in the quarter and penny landing on heads given that the quarter
> ***landed*** on heads?
>
>
> O = {qh ph, qh pt, qt ph qt pt)
>
> so we are looking for
>
> P(penny = heads | quarter = heads) ??
>
> we see that there are 2 possible choices where the quarter = heads and only
> one of them has the penny as heads also.. hence the probability is 1/2
>
>
I agree with that. Now, Suppose that a quarter and a penny were
flipped, that they both landed tails. Then, Bruce was told, "Two coins
were tossed and at least one was a tails. What are the chances for two
tails?" Bruce bet for two tails. Bob was told, "Two coins were tossed
and the penny landed tails. What are the chances for two tails?" Bob
bet for two tails.
Bob and Bruce will both win. Should they get different odds? They are
both betting on the same coin flip. If the coins were flipped three
ways, they should both get two to one. If the coins were flipped four
ways, they each bet even.
There is not much argument that Bob should bet even. That should take
the ambiguity out of Bruce's bet.
>
> now turn the penny into a quarter, which doesn't ofcourse effect the
> probability.
>
> Your issue then is obivously one of distinguishability. If you distingish
> the two quarters then it translates into "probability that FIRST quarter
> flipped is a head" as compared to the indistingishable case "probability
> that either the FIRST OR SECOND flipped is a head"!!!! Thats a huge
> difference.
>
The answer doesn't depend upon distinguishability. It depends upon
whether the coins were flipped four ways, or three. To flip two coins
three ways, the null must be selected, prior to the toss. Prior
selection is the key, not distinguishability.
>
> i.e., your problem as you have stated isn't "atleast one is head" but
> "atleast one is heads" and "atleast the first is heads" but since you leave
> out the "first". How you are doing this, it seems, is that you are changing
> the "rules" after the first flip i.e., switching it from unordered to
> ordered(or vice versa) after the first flip to arrive at a different answer.
>
That is not my problem. I don't think. If it is, there are some typo's.
> again, you seem to be saying that
>
> "given atleast one" means that "the first one"
>
> and
>
> "atleast one" means "either first or second"
>
No, no, no, no, no, no, no. "At least one is" means exactly what it
says. "Given at least one" has special mathematical meaning. It is
precisely defined by a little mathematical formula.
Two coins were tossed. We haven't looked. We know that the probability
for hh =1/4,ht=1/4,th=1/4,tt=1/4.
WE can say "given at least one head, the probability for two heads is
1/3."
We can say "given at least one tail, the probability for two tails is
1/3."
We can make those two statements, prior to any knowledge of the
outcome.
Bruce can make two bets, one on each statement. He'll win one bet, or
lose two, therefore he should get two to one.
We can't make the "at least one is" statements, as true statements,
without examining at least one coin.
> you think that because we say "given" we must have flipped the first already
> and hence we "ignore" its probability because it already happend and there
> is no reason to take it into account... a temporal problem.
>
No, no, no, no. I don't think that.
> your problem is perfectly valid to some extend but erroneous in the sense
> that you are taking into account something that you shouldn't. You are
> misinterpreting the semantics of the sentence.
>
This is a semantics problem, and the semantics is on my side. Two coins
were flipped, it has already happened. At least one is a tail. At least
one coin must have been inspected. At least one is a head, at least
one coin must have been inspected.
What does it mean to say "at least one is a tail"? It does not mean
that we reflipped at hh. Given at least one head means that we have
some kind of null at tt.
Semantics? I don't misinterpret semantics.
> i.e.
>
> hopefully this gets directly at the problem.
>
> I flip a quarter and it lands and I see the the actual side of the quarter
> but you have not seen it. I ask you "What is the probability that the
> quarter is on heads" and you answer 1/2... I SAY "NO YOU IDIOT, ITS 0
> BECAUSE IT LANDED ON TAILS!!!!"... and you look at me like I'm retarded
> because you know that there is a 1 out of 2 chance it would have landed on
> tails.
>
I'm not the one having trouble with semantics.
>
> i.e.
>
> P(heads | heads) != P(heads)
>
> (your problem is just a step away from this by added another variable but
> the issue is basicaly the same so it looks more rational than the above).
>
> As they say, You are comparing apples and oranges(posterior and priori
> probability)
>
> Anyways, thats my guess.
>
If you'll pay close attention to what I've told you, and understand it,
then you won't have to guess.
Eldon:)
> Jon
Gerry Myerson
> If I asked someone "what is 2 + 2"? they would naturally
> reply "4". If I then said "Wrong, nitwit, my 'policy' is
> to only ask questions whose answer is to be calculated
> in base 3; so the answer is 1", they would be justified
> in thinking _I_ was the nitwit for not including this
> proviso in the statement of the question!
When you say, "base 3," do you really mean to say something
like "modulo 3"?
In base 3, the answer to 2 + 2 is 11, which is pronounced "four".
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
quasi
>Two coins were flipped until at least one is a head. The probability
>for two heads is 1/3.(the until tells us that there was preselection,
>the writer will reflip at tt. In order to do so the writer must inspect
>both coins)
Right.
>Two coins were flipped, at least one is a tail. The probability for two
>tails is 1/2. (no preselection, no evidence that the writer of the
>statement has inspected more than one coin)
Wrong. The probability is still 1/3.
There was a preselection -- nature preselected.
But it still has to be a random flip, or else any bias would need to
be specified in the problem, so the coins are equally likely to be th,
ht, tt.
Also, it doesn't matter whether the writer inspected both coins.
If the first coin inspected is a tail, there is no need to look at the
second coin. The writer can simply stop and announce: "At least one
coin is a tail". So in this case, if the reader knew that only one
coin was inspected, the probability of tt would be 1/2.
However if the first coin inspected is a head, then it is necessary
for the write to look at the second coin. If we are assuming, by
preselection of nature, that the second coin is a tail, then after
looking at the second coin, again the writer announces: "At least one
coin is a tail". Now in this case, if the reader knew that the writer
was forced to look at 2 coins, the probability of tt would be 0.
If you look at those 2 probabilities, 1/2 and 0 -- it should make it
obvious that if you don't know how many coins were inspected, then the
probability of tt should be strictly less than 1/2.
The fact that it's 1/3 and not 1/4 is because the 2 cases above are
not equally likely. 2/3 of the time, only 1 coin to needs to be
inspected and 1/3 of the time, the writer will be forced to inspect
both.
Thus the probability of tt can be calculated as (2/3)*(1/2)+(1/3)*0
which gives an answer of 1/3.
quasi
john_r...@sagitta-ps.com
Gerry Myerson wrote:
>
> In article <1128888309....@g43g2000cwa.googlegroups.com>,
> john_r...@sagitta-ps.com wrote:
>
> > If I asked someone "what is 2 + 2"? they would naturally
> > reply "4". If I then said "Wrong, nitwit, my 'policy' is
> > to only ask questions whose answer is to be calculated
> > in base 3; so the answer is 1", they would be justified
> > in thinking _I_ was the nitwit for not including this
> > proviso in the statement of the question!
>
> When you say, "base 3," do you really mean to say something
> like "modulo 3"?
Yes, I meant mod 3
David C. Ullrich
>
>ma...@mimosa.csv.warwick.ac.uk wrote:
>>
>> In article <1128873353.7...@o13g2000cwo.googlegroups.com>,
>> john_r...@sagitta-ps.com writes:
>> >
>> >Christian Bau wrote:
>> >>
>> >> In article <1128860336.2...@f14g2000cwb.googlegroups.com>,
>> >> john_r...@sagitta-ps.com wrote:
>> >>
>> >> > quasi wrote:
>> >> > >
>> >> > > If information is "given", how is it given? Someone tells you,
>> >> > > right? I can't think of any other way to interpret "given".
>> >> >
>> >> > Exactly, and the natural interpretation is that this condition
>> >> > is being given by the person stating the question. Even if not,
>> >> > one must assume that the information is equally accurate. So,
>> >> > although there may be a hairsplitting logical distinction in
>> >> > how the question is framed, the salient facts are the same.
>> >>
>> >> Lets say two coins have been thrown, I can't see the result, but you
>> >> can. I want to find out a bit more. I ask a question and you have to
>> >> answer thruthfully.
>> >>
>> >> Case 1: I ask "Is at least one coin heads?", and the only possible
>> >> statements that you can make are "None of the coins is heads" and "at
>> >> least one of the coins is heads". This is the same as "... given that
>> >> ..."
>> >>
>> >> Case 2: I tell you: "Please give me some true statement about the
>> >> coins". Now you have an infinite amount of choices. This is the
>> >> same as "... somebody told me that..."
>> >
>> >But once I've chosen to tell you truthfully that there's at least
>> >one head, your state of knowledge is the same as it is after my
>> >reply to case 1 - Different route, same destination which, in
>> >the context of the problem, you have already reached.
>>
>> I don't agree that your state of knowledge is the same after getting the
>> same replies in Cases 1 and 2. In Case 1 there is no ambiguity at all.
>> If the I reply "at least one coin is heads" then the proabbility of two
>> heads is 1/3.
>>
>> But in Case 2, you have no idea what policy I adopted in choosing
>> which statement to make.
>>
>> [...]
>
>I understand what you're saying, and it's obviously right
>if one takes policies and hidden motives into account.
>But, like Christian B with his incentives and lies, in
>just mentioning the word "policy" you're extending the
>problem beyond its original statement, reading between
>lines to introduce aspects which (by convention, we can
>assume) aren't there unless explicitly mentioned.
And in particular, once we introduce such factors
then we cannot possibly say anything about what the
answer to the question is. _If_ we're considering
questions of who said what and why they would say
that, but we're given no information about all that,
then saying that it follows that the probability is
anything specific is just stupid.
>If I asked someone "what is 2 + 2"? they would naturally
>reply "4". If I then said "Wrong, nitwit, my 'policy' is
>to only ask questions whose answer is to be calculated
>in base 3; so the answer is 1", they would be justified
>in thinking _I_ was the nitwit for not including this
>proviso in the statement of the question!
>
>In fact thinking about it some more, it seems to me
>that "X given Y" can quite reasonably be interpreted
>as "X assuming Y", even if in principle there is this
>meta-statement aspect to the first form.
************************
David C. Ullrich
Markus Sigg
> If I asked someone "what is 2 + 2"? they would naturally
> reply "4". If I then said "Wrong, nitwit, my 'policy' is
> to only ask questions whose answer is to be calculated
> in base 3; so the answer is 1", they would be justified
> in thinking _I_ was the nitwit for not including this
> proviso in the statement of the question!
If you live in a mod 3 world, then don't call 1 a correct answer
and 4 a wrong answer, because in your world they are equal.
Markus
John O'Flaherty
Elmo wrote:
> Conditional probability is
> P[A|B] = P[AB]/P[B].
>
> In this formula, P(B) is defined as the event which has definitely
> happened.
>
> When we get 3/4 in the numerator, the answer is 1/3.
That would happen with 3/4 in the denominator, not the numerator.
> This defines an
> event which has definitely happened, or will definitely happened. It
> precisely defines what it means to say "given at least one." It also
> precisely defines a coin flip sequence. There is a precise sequence of
> events which must happen to get this result.
There is no question of a sequence in the problem as I understood it.
Two coins have been flipped, it matters not if simultaneously. What you
know about the outcome of a total of two flips is that there is at
least one head. That is given.
> Alter the numbers in the denominator and get a different answer. When
> the denominator equals 1/2, the answer to our question is 1/2. This
> precisely defines a different coin flip sequence.
>
> A coin was flipped, it landed heads. I would not assume that heads was
> chosen, then the coin was flipped.
When you talk about 'chosen', I guess you are talking about an entirely
different problem, something to do with an individual calling 'heads',
flipping a coin and seeing what the outcome is. Is that what you mean?
> Two coins were flipped and at least one landed heads. I would not
> assume that heads were chosen, then the coins were flipped. When two
> coins were flipped and they landed tails, tails, "at least one is a
> tails" is a true statement. When two coins were flipped and they landed
> heads, heads, "at least one is a head" is a true statement.
Again, what do you mean by 'chosen'?
> Suppose that we decide to flip for a heads, ie, we will say at least
> one heads at hh, ht, and th.
What do you mean by 'flip for a heads'? If you mean predicting an
outcome, that's a whole different problem.
> Then, suppose that the coins land tt, and we say, "two coins were
> flipped and at least one is a tail." That is a true statement. My
> supposition is that "given at least one tail" means something different
> from that.
"Given at least one tail" means exactly that there was at least one
tail. The relevant meaning of given, from American Heritage Dictionary-
b. Granted as a supposition; acknowledged or assumed.
In the case of two coin flips, when it is 'given' that there was at
least one head, it means the possible outcomes are HT, TH, and HH. The
possibility of HH, which was 1/4 of the original array of TT, HT, TH,
and HH, is now one third of the remaining possibilities, after TT has
been excluded.
--
john
Shmuel (Seymour J.) Metz
at 01:27 PM, jdo...@math-cl-n03.math.ucr.edu (James Dolan) said:
>sorry, your question is just too stupid to bother answering.
IOW, you don't know the answer.
*PLONK*
Wayne Brown
> On 8 Oct 2005 07:12:35 -0700, "Elmo" <elmo...@yahoo.com> wrote:
>
>>[...]
>>
>>Dr. Ullrich seems to be somewhat of a fakir. He plays a lot of
>>bachgammon.
>
> That's "faker", and "backgammon".
Drat. I was picturing you wearing a turban and walking over hot coals, or
sleeping on a bed of nails, or climbing a rope into the sky, and perhaps
inventing a new game involving 18th-century composers. sci.math can be
*so* disillusioning...
--
Wayne Brown (HPCC #1104) | "When your tail's in a crack, you improvise
fwb...@bellsouth.net | if you're good enough. Otherwise you give
| your pelt to the trapper."
e^(i*pi) + 1 = 0 -- Euler | -- John Myers Myers, "Silverlock"
Elmo
> Elmo wrote:
> > Conditional probability is
> > P[A|B] = P[AB]/P[B].
> >
> > In this formula, P(B) is defined as the event which has definitely
> > happened.
> >
> > When we get 3/4 in the numerator, the answer is 1/3.
>
> That would happen with 3/4 in the denominator, not the numerator.
>
> > This defines an
> > event which has definitely happened, or will definitely happened. It
> > precisely defines what it means to say "given at least one." It also
> > precisely defines a coin flip sequence. There is a precise sequence of
> > events which must happen to get this result.
>
> There is no question of a sequence in the problem as I understood it.
> Two coins have been flipped, it matters not if simultaneously. What you
> know about the outcome of a total of two flips is that there is at
>
choose heads and flip for one?
> > Alter the numbers in the denominator and get a different answer. When
> > the denominator equals 1/2, the answer to our question is 1/2. This
> > precisely defines a different coin flip sequence.
> >
> > A coin was flipped, it landed heads. I would not assume that heads was
> > chosen, then the coin was flipped.
>
> When you talk about 'chosen', I guess you are talking about an entirely
> different problem, something to do with an individual calling 'heads',
> flipping a coin and seeing what the outcome is. Is that what you mean?
>
At least one is a head, heads were chose, at least one is a tail, tails
were chosen, given at least one tail, tails were chosen.
> > Two coins were flipped and at least one landed heads. I would not
> > assume that heads were chosen, then the coins were flipped. When two
> > coins were flipped and they landed tails, tails, "at least one is a
> > tails" is a true statement. When two coins were flipped and they landed
> > heads, heads, "at least one is a head" is a true statement.
>
> Again, what do you mean by 'chosen'?
>
> > Suppose that we decide to flip for a heads, ie, we will say at least
> > one heads at hh, ht, and th.
>
> What do you mean by 'flip for a heads'? If you mean predicting an
> outcome, that's a whole different problem.
>
That's fairly self explanatory. Were the coins flipped and an outcome
reported, or did the flipper decide upon heads prior to the flip? Call
this flipped with extreme prejudice toward heads, which would be
extreme prejudice against tails.
> > Then, suppose that the coins land tt, and we say, "two coins were
> > flipped and at least one is a tail." That is a true statement. My
> > supposition is that "given at least one tail" means something different
> > from that.
>
> "Given at least one tail" means exactly that there was at least one
> tail. The relevant meaning of given, from American Heritage Dictionary-
> b. Granted as a supposition; acknowledged or assumed.
> In the case of two coin flips, when it is 'given' that there was at
> least one head, it means the possible outcomes are HT, TH, and HH. The
> possibility of HH, which was 1/4 of the original array of TT, HT, TH,
> and HH, is now one third of the remaining possibilities, after TT has
> been excluded.
>
Yes, two coins flip four equally likely ways. When two coins are
flipped then TT, HT, TH should each happen 1/4 of the time, therefore
they represent 3/4 of the total. If the coins were flipped as four,
then, after inspection the "at least one is a head" was made, the three
are all that is left, however, they are no longer equally likely.
To flip two coins three equally likely ways, the null must have been
decided, prior to inspection.
The probability for two heads, given at least one, is derived in the
conditional probability formula. It is derived with extreme prejudice
toward heads. This means that, prior to the inspection, we can say,
"Two coins were flipped. Given at least one head, the probability for
two heads is 1/3.
Given at least one tail, the probability for two tails is 1/3.
Eldon
Eldon
> --
> john
ste...@nomail.com
> John O'Flaherty wrote:
>>
>> There is no question of a sequence in the problem as I understood it.
>> Two coins have been flipped, it matters not if simultaneously. What you
>> know about the outcome of a total of two flips is that there is at
>>
> Did the flipper flip the coins and discover a head? Or, did the flipper
> choose heads and flip for one?
Read what he wrote.
>> Two coins have been flipped, it matters not if simultaneously.
Two coins have been flipped.
Stephen
mark mcsweeney
Jim Spriggs
>
> On this forum I have argued the question, "Two coins were flipped and
> at least one is a head. What are the chances that there are two heads?"
>
> Many mathematicians get it confused with "The probability for two
> heads, given at least one head?"
>
> I say that "given at least one head", and told, "at least one head"
> mean two different things.
You could always write a computer simulation. Perhaps that would
convince you.
--
I don't know who you are Sir, or where you come from,
but you've done me a power of good.
Elmo
No, no. There is a head on one side of the coins, a tail on the other.
I never have denied that.
Eldon
Richard Tobin
Jim Spriggs <jim.s...@ANTISPAMbtinternet.com.invalid> wrote:
>> I say that "given at least one head", and told, "at least one head"
>> mean two different things.
>You could always write a computer simulation. Perhaps that would
>convince you.
More to the point, it will force him to make explicit what he means
by the phrases.
-- Richard
ma...@mimosa.csv.warwick.ac.uk
Perhaps he means that "given at least one head" is specifying an
abstract mathematical problem in probability theory, whereas "told
at least one head" is describing an actual event.
So, imagine you are walking along the street minding your own business
one day, when a stranger accosts you and announces "I have just tossed
two fair coins, and at least one of them landed heads!". We happen to
know somehow that this stranger always tells the truth, but otherwise we
know nothing about him or his motivations. In that situation, can we say
anything meaningful at all about the probability of two heads?
Derek Holt.
Elmo
I agree with that, but let's leave out the part about meeting the man
on the street. All we have is a statement which says, "Two coins were
flipped and at least one of them landed heads." We don't know who said
it, it might have been computer generated. We want to know what we can
say about the coin flip.
We can say:
1.Two coins were flipped. We know because our statement told us so.
2.All we know about the coin flip, we learned from the statement.
3.We know that TT did not happen.
4.We know that HH happened, and the statement was made, or, HT happened
and the statement was made, or, TH happened and the statement was made.
5.Two coins were tossed is a statement of fact.
6."At least one is a head" is a conditional statement.
7.Prior to the conditional statement HT and HH were equally likely. Two
of four equally likely events.
8.Now, they are two of three remaining events. Are they still equally
likely?
9.Were the coins flipped three equally likely ways?
10.How do you flip two coins three equally likely ways?
11.There are methods to flip two coins three ways. Tossing them, then
making an "at least one is" statement ain't one of those methods.
12.Dr. Ullrich should answer this. How would he toss three coins three
equally likely ways? To my knowledge he has not answered that.
Hello again, Dr. Holt. How would you flip two coins three ways?
Thanks for your input,
Eldon Moritz
Elmo
were flipped and at least one is a head."
Two coins ordinarilly flip four equally likely ways. There are methods
for flipping two coins three equally likely ways? Was this one of them?
How would you flip two coins three equally likely ways?
Eldon
Keith A. Lewis
>So, let's not make stupid statements. We have the statement, "Two coins
>were flipped and at least one is a head."
>
>Two coins ordinarilly flip four equally likely ways. There are methods
>for flipping two coins three equally likely ways? Was this one of them?
>
>How would you flip two coins three equally likely ways?
One standard method is to flip two (different looking) coins, until you get
a result which is not TT. Then the probabilities of HH, TH, and HT are all
equal (assuming the coins are fair).
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
John O'Flaherty
If you are talking about finding one coin was heads, taking it away and
figuring the chances on the second coin, that is a different problem.
It is a problem in which the state of the second coin is entirely
divorced from that of the first. It doesn't correspond to the usual
meaning of 'given that one is a heads'. When that statement is made,
you don't know which is the coin that was heads, and all three possible
outcomes are equally likely.
> The probability for two heads, given at least one, is derived in the
> conditional probability formula. It is derived with extreme prejudice
> toward heads. This means that, prior to the inspection, we can say,
> "Two coins were flipped. Given at least one head, the probability for
> two heads is 1/3.
> Given at least one tail, the probability for two tails is 1/3.
Well, then you've finally seen it... right?
--
john
Elmo
Go to the head of the class.
Select a null, prior to the flip, that's the key.
Two coins were flipped, until at least one was a head.
Two coins were flipped, until they were not HH.
Two coins were flipped, until they were not HT.
Two coins were flipped, until they were not TH.
For the last two to work the two coins must be in some way
distinguishable.
The 'until' tells us that the null was selected prior.
Eldon:)
Elmo
least 1) It's defined mathematically, and I don't argue with the math.
What I've argued is that when two coins are flipped, and an "at least
one is" statement is made about them, it takes different numbers in the
formula to correctly decifer what happened.
Read what I've been writing. I haven't finally seen it.....right? I
know as much, or more about this one question than anyone living. Try
to understand what I've been writing......okay?
Eldon
> --
> john
Dave Rusin
<ma...@mimosa.csv.warwick.ac.uk> wrote:
>So, imagine you are walking along the street minding your own business
>one day, when a stranger accosts you and announces "I have just tossed
>two fair coins, and at least one of them landed heads!". We happen to
>know somehow that this stranger always tells the truth, but otherwise we
>know nothing about him or his motivations. In that situation, can we say
>anything meaningful at all about the probability of two heads?
Ordinarily we could say that "they are better than one". But when one
of the two heads belongs to someone who is accosting strangers with
bizarre tales of flying coins, I think the probability is that these
two heads are NOT better than one.
dave
sugna...@gmail.com
Elmo wrote:
> On this forum I have argued the question, "Two coins were flipped and
> at least one is a head. What are the chances that there are two heads?"
Would the following experiment meet with your approval? If not, then
why not?
1. Take a thousand pairs of coins (2000 coins in total)
2. Toss each pair of coins *once only*
3. Remove all pairs that were tail-tail
4. Count the total number of remaining pairs, and call this N(total)
5. Count the number of remaining pairs that are h-h, and call this
N(h-h)
6. Calculate N(h-h)/N(total), and call this p(h-h)
7. Define p(h-h) as the answer to the question: "Two coins were flipped
and at least one is a head. What are the chances that there are two
heads?"
If you accept that this meets your criteria, then it will be very easy
to settle the discussion with a real-coin or computerized simulation;
if you don't accept it, could you please propose your own experiment in
equivalent terms?
Jesse F. Hughes
> We can say:
> 1.Two coins were flipped. We know because our statement told us so.
> 2.All we know about the coin flip, we learned from the statement.
> 3.We know that TT did not happen.
> 4.We know that HH happened, and the statement was made, or, HT happened
> and the statement was made, or, TH happened and the statement was made.
> 5.Two coins were tossed is a statement of fact.
> 6."At least one is a head" is a conditional statement.
I have held my tongue until now, but in what reasonable sense is "at
least one is a head" a conditional statement? It is no less factual
than "two coins were tossed."
Utterly bizarre.
Conditional statements are statements of the form: If X then Y. Or Y,
given X. Or Y only if X. Or....
"At least one is a head" is not a stinking conditional statement.
--
Jesse F. Hughes
"That's what's brutal about mathematics! When you're wrong, you can
have spent years, and lots of effort, and come out at the end with
nothing." -- James S. Harris on the path of self-discovery (?)
Randy Poe
Jesse F. Hughes wrote:
> "Elmo" <elmo...@yahoo.com> writes:
>
> > We can say:
> > 1.Two coins were flipped. We know because our statement told us so.
> > 2.All we know about the coin flip, we learned from the statement.
> > 3.We know that TT did not happen.
> > 4.We know that HH happened, and the statement was made, or, HT happened
> > and the statement was made, or, TH happened and the statement was made.
> > 5.Two coins were tossed is a statement of fact.
> > 6."At least one is a head" is a conditional statement.
>
> I have held my tongue until now, but in what reasonable sense is "at
> least one is a head" a conditional statement? It is no less factual
> than "two coins were tossed."
>
> Utterly bizarre.
>
> Conditional statements are statements of the form: If X then Y. Or Y,
> given X. Or Y only if X. Or....
>
> "At least one is a head" is not a stinking conditional statement.
Eldon has been obsessing about this probability calculation
for years.
See, for instance, this page from 1998:
http://www.wiskit.com/marilyn/boys.html
I think his answer would be something like "You don't know
the conditions under which the person would tell you that
at least one is a head. In the case HT/TH, they might
be equally likely to say that one is a tail."
He has severe problems with the idea of "given" and insists
on constructing elaborate behavioral scenarios involving
the means by which you obtain the information that at
least one is a head.
- Randy
Elmo
> equivalent terms?
If you will read a little further, we are examining the difference
between, "Two coins were flipped and at least one is a head", and "Two
coins were flipped, given that there is at least one head." I will
accept your program for "given at least one head." It's easy to see
that it will give 1/3.
Dr. Ullrich said:
Exactly what is the difference in meaning between
"Two coins were flipped and at least one is a head. What
are the chances that there are two heads?"
and
"The probability for two heads, given at least one head?"
?
<End of Quote>
Eldon's examination starts here>
For our examination, use the following two questions, Q1, and Q2.
Q1.Two coins were flipped and at least one is a head. What
is the probability that there are two heads?"
Q2.Two coins were flipped and given at least one head. What
is the probability that there are two heads?"
1.Either they are the same, or they are not.
2.The only difference between Q1, and Q2 are the terms, "at least one
is" and "given at least one". If they have different probabilities, the
difference was caused by using, or not using "given".
3.If they are the same, then the two terms are interchangeable. Let's
assume that they are the same.
4.Toss forty thousand pairs of coins. Hypothetically we could have ten
thousand each of HH, HT, TH, and TT. Any one of the forty thousand
could have been the first toss.
5. There is at least one true, "at least one is" statement for every
toss. As we have one statement, one toss, let's make one statement per
toss. At HH, "at least one is a head", at TT, "At least one is a tail".
At HT and TH, either statement would be true. Make either one without
preference.
6.Now substitute for "at least one is", "given at least one".
7.We have incongruity somewhere, can you find it? Maybe, just maybe,
the two terms don't interchange.
8.I say that "given at least one" is the special case whereas, "at
least one is a head" is stated with extreme prejudice toward heads, or,
"at least one is a tail" is stated with extreme prejudice toward tails.
9.State "at least one is a head" with extreme prejudice toward tails,
and the probability for two heads would be one.
10. State "at least one is a head" without prejudice and the answer is
1/2.
11. As we can state "at least one is", with, or without prejudice,
"given at least one" has special meaning, and is not interchangeable
with "at least one is".
Eldon:)
john_r...@sagitta-ps.com
Randy Poe wrote:
>
> Eldon has been obsessing about this probability calculation
> for years.
>
> See, for instance, this page from 1998:
> http://www.wiskit.com/marilyn/boys.html
>
> I think his answer would be something like "You don't know
> the conditions under which the person would tell you that
> at least one is a head. In the case HT/TH, they might
> be equally likely to say that one is a tail."
>
> He has severe problems with the idea of "given" and insists
> on constructing elaborate behavioral scenarios involving
> the means by which you obtain the information that at
> least one is a head.
On the evidence of some replies in this thread, it looks
like he's not the only one with that problem.
In fact it almost seems as if the "take it as you find it"
camp has been in the minority!
Elmo
Randy, I have no problem with "given". I have a problem with people who
see the term "at least one" and rush to the formula for "given at least
one."
I've quit this question several times, but people send me emails about
it, and I get going again. I spend a lot of time hanging around
airports, with internet access. Through the process, I've gotten to be
the world's foremost authority on this question. If you think I don't
understand this question, read and understand what I've written on it.
Eldon
Elmo
Jesse F. Hughes wrote:
> "Elmo" <elmo...@yahoo.com> writes:
>
> > We can say:
> > 1.Two coins were flipped. We know because our statement told us so.
> > 2.All we know about the coin flip, we learned from the statement.
> > 3.We know that TT did not happen.
> > 4.We know that HH happened, and the statement was made, or, HT happened
> > and the statement was made, or, TH happened and the statement was made.
> > 5.Two coins were tossed is a statement of fact.
> > 6."At least one is a head" is a conditional statement.
>
> I have held my tongue until now, but in what reasonable sense is "at
> least one is a head" a conditional statement? It is no less factual
> than "two coins were tossed."
>
> Utterly bizarre.
>
> Conditional statements are statements of the form: If X then Y. Or Y,
> given X. Or Y only if X. Or....
>
> "At least one is a head" is not a stinking conditional statement.
>
I am not a mathematician. I have about 25 hours of college math. I am
an intelligent person who has spent a lot of time on this one question.
Dr. H.L. Gray, chaired Professor of Mathematical Statistics at SMU in
Dallas, told me that "two coins were tossed" is a statement of fact. He
also said that, "at least one is a head" is a conditional statement.
We know that two coins were tossed because the statement told us so.
It's a statement of fact.
On the condition that we were told "at least one is a head" we wish to
know if HH, and HT are still equally likely. Sounds like a stinking
conditional statement to me.
Even though I'm not a mathematician, Dr. Gray keeps from running around
completely without adult supervision.
Suppose that two coins were tossed and they landed TT. The statement
was generated, "Two coins were tossed and at least one is a tail."
Bruce only heard the statement, he bet one token for two tails, he will
win. What odds should he collect?
Randy is wrong, I don't know, or care who made the statement. I only
know that the statement exists, and it constitutes the entire question.
Eldon
Elmo
> In article <1128899117.6...@g44g2000cwa.googlegroups.com> "Elmo" <elmo...@yahoo.com> writes:
> ...
> > Given at least one heads means that we'll have a success every time at
> > hh, ht, and th. At least one is a head means whatever it says.
>
> No, that is not what is implied. It is implied that the current toss
> has at least one head. Nothing more, nothing less. But you are talking
> about semantics rather than mathematics. You assume that, in some way,
> the sentence "given one is head" means that it would also be possible
> that the sentence would have been "given one is tail". Or in one of it's
> many disguises.
>
I assume that heads and tails are equally likely.
I assume that the problem statement is true.
I know that "two coins were tossed, given at least one head, and given
at least one tail," have the same probability for two of the same.
When two coins were tossed, they can land four equally likely ways. Any
one way could have landed first, therefore we can speak of each of them
individually as though they were the first toss.
Suppose that:
Q1. TT landed first. The statement was generated, "Two coins were
tossed and at least one is a tail. What is the probability for two
tails?" Bruce only heard the statement. He bet one token for TT, and he
will win. What odds should he collect.
Or,
Q2. HH landed first. The statement was generated, "Two coins were
tossed and at least one is a head. What is the probability for two
heads?" Bruce only heard the statement, he bet one token for HH.He will
win, what odds should he collect.
I don't assume, I know that Bruce should collect the same odds at Q1,
or Q2.
Q3. Suppose that at TT, Bob was told "Two coins were tossed and the
dime landed tails. What is the probability for two tails? Bob only
heard the statement, and bet for TT. He will win, should he collect
different odds from Bruce? I don't assume, I know that he should not
collect different odds.
> Good. Let me formulate it differently. Two coins are tossed, and it
> is announced that one is either heads or tails. What is the probability
> that the other is the same? The probability is 1/2.
>
I agree with this scenario.
> Another formulation. Two coins are tossed and it is announced that one
> is heads, or nothing is announced. What is the probability that both are
> equal? Assuming that there is an announcement when at least one head
> crops up, the probability is again, 1/2.
>
I believe that this one is 1/3, but I may not understand exactly what
you're saying.
> A final formulation. Two coins are tossed and it is announced that one
> is heads, or nothing is announced. Assuming that there is an announcement
> when at least one head crops up, what is the probability that, given such
> an announcement, both are the same? 1/3.
I agree with this scenario.
When a scenario is defined, then it is usually not difficult to get a
correct answer. Our job, however is not to describe a scenario, and
therefore a different question which we can answer. Our job is to
answer the question, as written.
We know that the coins were tossed and a statement was made. We have a
one to one ratio of statements to tosses. So long as that ratio stands,
the bettor for two of the same should win half the time. For the bettor
for one of each to win two thirds of the time, the statement should
show a three statements to four tosses ratio.
Example:
Two coins were tossed until at least one is a head. (three statements
out of four, and some kind of demur, pay the winner two to one at HH)
Eldon
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Randy Poe
Elmo wrote:
> Randy, I have no problem with "given". I have a problem with people who
> see the term "at least one" and rush to the formula for "given at least
> one."
And I'm in the camp that they say exactly the same thing, and
can not figure out how interpreting them as the same is
"rushing" to anything. It's like "rushing to the conclusion"
that somebody concludes Joe is dead just because somebody
says he died yesterday.
- Randy