Marilyn von Plutonium on Fermat
Benjamin J. Tilly
tyc...@cayley.mit.edu (Timothy Y. Chow) writes:
(Conversation on how well MVS does deleted.)
> Someone else I know claims that she analyzed the two envelopes problem
> incorrectly, but I am not familiar with the column in question.
She did get it wrong. However her basic intuition was the same one that
most mathematicians would have. It just turns out that the answer is
nonintuitive. The problem under consideration here is the one where you
have two envelopes, one of which has twice as much money. You are given
one randomly and you know how much is in that one. Should you switch?
In fact there is a strategy about switching which will have better than
even odds of having you get the larger, but the analysis is somewhat
subtle, and you will have no idea of how much better than even your
odds are. The situation in which that strategy works (just the two
envelopes, but you do not need one to have twice the money, just two
different numbers...) was discussed ad nauseum on sci.math over the
summer.
Ben Tilly
Jim Propp
of other people are also planning on writing letters, some more temperate
than others. I would just want to caution people that it's entirely possible
that Ms. vos Savant will quote selectively from such letters and make us
mathematicians out to be quite an arrogant bunch. For instance, if you write
a letter that gives a well-reasoned argument and then concludes with a
sentence like "I think it highly indefensible for you to have done such a
shoddy job," you could very well find yourself credited with the closing
sentiment but not with the logic that backs it up. And Ms. vos Savant
could airily comment on your letter with an amused remark along the lines of
"Isn't it interesting how hostile academics can get when their sacred cows
are attacked?"
Ms. vos Savant is wrong, but she does at least manage to be civil. Let's
show that we can match her in civility even as we challenge her under-informed
convictions. We'll be doing the profession a favor.
Jim Propp
Department of Mathematics
Massachusetts Institute of Technology
Mr. Flibble
Jim Propp <pr...@godel.mit.edu> wrote:
[Cautions about the possibility of being quoted out of context deleted.]
>
>Ms. vos Savant is wrong, but she does at least manage to be civil.
I actually disagree with this. She manages to be very uncivil in
a very underhanded way. The entire article contained little snide comments
like "That might seem proof enough for the general public, but for
mathematicians it's no proof at all." (Which I read as implying that
mathematicians are out of touch with good old common sense that the
general public has.) Also, her comments about Wiles are very snide (something
she seems to be very good at being), implying that he has become so
obsessed with this that he's neglected his wife and kids and let the
house go to seed.
The whole thing seems to have a tone of "Oh, of course I know
better than all these people." I found that far more annoying than the
arguments she was making. I can deal with ignorance. Arrogance is a lot
harder to swallow.
All of this is my reading between the lines of the article of
course. Your mileage may (and probably will) vary.
-Jeff
--
Jeff Hildebrand, The Shaggy TA hild...@math.wisc.edu
"With the end of the Cold War, is homotopy theory really the next big
threat to your security?" - political theory from Don Shimamoto
Benjamin J. Tilly
Since I was asked about it, here is the solution. First of all the
problem is the following. There are two envelopes with different
numbers in them. I get one at random and look at the number in it. With
no more information I can make an educated guess that I *know* has
better than even odds of being right as to which one has the larger
number in it.
My strategy is to pick a number from some specified continuous
distribution with non-zero density everywhere, such as the standard
normal distribution, and pretend that that number is the number in the
other envelope. Now let us analyze that. First of all since the
distribution is continuous I know that I will pick one of the numbers
in the two envelopes with probability 0. Therefore we can disregard
that. If the number is bigger than both of the ones in the envelopes
then I will think that I have been given the smaller, and there are
even odds that I am right. Similarly my odds are even if that number
was smaller than both of the ones in the envelopes. But if it is
*between* the two in the envelopes then I _will_ get it right! Working
that out it turns out that my odds of being right are
0.5 + 0.5 P{the number chosen is between the numbers in the envelopes}
But since the distribution has non-zero density everywhere that means
that I have better than even odds of being right.
However there is a subtle point here. The probability that I worked out
is the probability that I am right *given* the numbers in the
envelopes. Since I specified in the problem that the envelopes existed,
with specified numbers, *before* the probability is defined, that is
OK. However I would not have any idea of what my odds are when I was
playing, and indeed they could be only _very_ slightly better than
even.
Incidentally the person who wrote me suggested that this should go into
the rec.puzzles archives. I leave that up to whomever is in charge of
them. But it is IMO a very interesting and counterintuitive result.
BTW I did *not* invent this solution, and indeed I know that it has
been around for a very long time.
Ben Tilly
Carl Witthoft
>>Ms. vos Savant is wrong, but she does at least manage to be civil.
>
> I actually disagree with this. She manages to be very uncivil in
>a very underhanded way. The entire article contained little snide comments
>like "That might seem proof enough for the general public, but for
>mathematicians it's no proof at all." (Which I read as implying that
>mathematicians are out of touch with good old common sense that the
>general public has.) Also, her comments about Wiles are very snide (something
I've missed all the startup to this thread due to the usual newsreader
goofups :=(. However, I disagree with Flibble's interpretation of
Savant's statement. (I've also missed her columns 'cause I didn't get
the SUnday paper)
The quote above clearly means that mathematicians are far more careful
and more rigorous in their analysis than the public at large is.
What's wrong with that? -- except that the "public" doesn't knwo
much about math.
Besides, she got MonteHall right, which puts her 'way above half the
net :=( .
--
Carl Witthoft @ Adaptive Optics Associates
ca...@aoa.utc.com 54 CambridgePark Drive, Cambridge,MA 02140 617-864-0201
"Eight ever; nine never."
Frank Pinto
>Ben Tilly
Ben, I applaud you
clap, clap, clap, clap,..........
regards
Frank Pinto
--
Frank Pinto (XICO)| The roots of education are bitter,
Mathematics, | but the fruit is sweet.
Computer Science &|
Philosophy | ARISTOTLE
Gareth Rees
|> Since I was asked about it, here is the solution. First of all the
|> problem is the following. There are two envelopes with different
|>
|> My strategy is to pick a number from some specified continuous
|> distribution with non-zero density everywhere, such as the standard
|> normal distribution, and pretend that that number is the number in the
|> I _will_ get it right! [...] Since the distribution has non-zero density
There's a simpler strategy. Pick some strictly monotonic function f
from the non-negative reals [0,+infinity) to [0,1). If you find money x
in the envelope you open, pick the other envelope with probability
1 - f(x).
So if the numbers in the envelopes are x and y (x > y), the amount of
money you expect to get is
0.5 * [f(x) * x + (1 - f(x)) * y + f(y) * y + (1 - f(y)) * x]
= 0.5 * [x + y + x * (f(x) - f(y)) + y * (f(y) - f(x))]
= 0.5 * [x + y + (f(x) - f(y)) * (x - y)]
but f(x) - f(y) and x - y are strictly positive (as f is strictly
monotonic) so this is strictly more than you get by choosing at random.
--
Gareth Rees
S
>I am planning on writing a letter to Marilyn vos Savant; I imagine a number
>of other people are also planning on writing letters, some more temperate
>than others. I would just want to caution people that it's entirely possible
>that Ms. vos Savant will quote selectively from such letters and make us
>mathematicians out to be quite an arrogant bunch.
>
>Ms. vos Savant is wrong, but she does at least manage to be civil. Let's
>show that we can match her in civility even as we challenge her under-informed
>convictions. We'll be doing the profession a favor.
>
May I say something serious? I thinks that MvS was just a Man_made
character, so was that Plutonium! If, someone could produce a charactor
like Wiles, why can't they produce inverse charactors to make everybody
to hate them for whatever the political means.
Think about it!
My suggestion iis, if an article or a book is so poorly written,
just don't read it. Dump it into the trash. Else, you would be
just wasting your time. Someone would love to see cats
and dogs fighting each others, or even kill each others so they
can retreat to get themselves covered.
David Karr
>Benjamin J. Tilly writes:
>|> My strategy is to pick a number from some specified continuous
>|> distribution with non-zero density everywhere, such as the standard
>|> normal distribution, and pretend that that number is the number in the
>|> other envelope [...]
>
>from the non-negative reals [0,+infinity) to [0,1). If you find money x
>in the envelope you open, pick the other envelope with probability
>1 - f(x).
You've just described a strategy that is very nearly identical to Ben
Tilly's---in fact his distribution function can be used directly as
your f(x) in which case you will always behave the same. To see this,
remember that he will pick the other envelope if his random variable
turns out greater than the money in the first envelope. If f(x) is
the probability distribution of his random variable (that is, the
integral of the density, not the density itself), and the envelope has
x dollars in it, he picks the other with probability 1 - f(x).
The only differences I see offhand are some minor conditions on f(x)
which don't seem to make all that much difference.
-- David A. Karr (ka...@cs.cornell.edu)
Ludwig Plutonium
hs...@sdcc13.ucsd.edu (S) writes:
> In article <1993Nov23....@galois.mit.edu> pr...@godel.mit.edu (Jim Propp) writes:
> >I am planning on writing a letter to Marilyn vos Savant; I imagine a number
> >of other people are also planning on writing letters, some more temperate
> >than others. I would just want to caution people that it's entirely possible
> >that Ms. vos Savant will quote selectively from such letters and make us
> >mathematicians out to be quite an arrogant bunch.
Now, I do not mind one bit if Marilyn publishes my name amongst
several math professors. I will furnish her all the wrong answers she
wants--- to the pancake problem, to the two letter problem, to the
Monty doors. To a science revolutionary, good publicity is good; bad
publicity is good. No publicity is bad. I don't mind one bit having my
name in Parade along with math professors. Please, Ben Tilly or someone
help me out by telling me when Marilyn wants a new fresh crop of bad
answers to her silly problems.
CALLING ON -Her-Royal-Majesty-IQ-Marilyn, the probability that the
pancake heavy with butter lands on its edge is 2/3.
Sincerely, Ludwig Plutonium
Benjamin J. Tilly
TMo...@massey.ac.nz (Terry Moore) writes:
> In article <CH084...@dartvax.dartmouth.edu>,
> Benjamin...@dartmouth.edu (Benjamin J. Tilly) wrote:
> >
> > In article <CGyo...@dartvax.dartmouth.edu>
> > Benjamin...@dartmouth.edu (Benjamin J. Tilly) writes:
> >> The problem under consideration here is the one where you
> > > have two envelopes, one of which has twice as much money. You are given
> > > one randomly and you know how much is in that one. Should you switch?
>
> > Since I was asked about it, here is the solution. First of all the
> > problem is the following. There are two envelopes with different
> > numbers in them. I get one at random and look at the number in it.
>
> One actually had twice as much as the other.
>
If you had left the rest of the original comment in then it might have
been more apparent in context that the second part is trying to answer
a generalization of the first. Therefore the problem that I stated in
the second one is a general situation that includes the first as a
special case (other than the trivial case of no money in either.:-)
> > My strategy is to pick a number from some specified continuous
> > distribution with non-zero density everywhere, such as the standard
> > normal distribution, and pretend that that number is the number in the
> > other envelope.
>
> It can't be unless it is either twice the known number or half
> of it.
>
It can in the general solution.
> No doubt you can modify the solution for the problem at hand,
> but this doesn't quite succeed.
>
Since the specific problem at the start is a special case of the
general problem, which the method works for, this does work as a
solution. In addition there is another solution, which is basically
equivalent to mine, which has been posted also.
> Hey! I don't want to start the thread again, just tidy up the
> argument and we can forget it.
I think that it is fine as it is.
Incidentally there was one person who wrote me saying that their bridge
club was talking about this who wanted some details sorted out.
Unfortunately I just got a message back that says that my e-mail
message could not get through to you. If you write me again then I will
try again.
Ben Tilly
Bionic Tapeworm
>Besides, she got MonteHall right, which puts her 'way above half the
>net :=( .
NO NO NO!!! *TWO THIRDS* of the 'net.
J R Partington
Very droll. The trouble with a net is that most of it consists of
empty space, and it's usually the empty space that replies.
JRP
--
Dr Jonathan R. Partington, School of Mathematics, University of Leeds,
Leeds LS2 9JT, U.K.
John Major's latest embarrassment -- the arms to IRA affair.
Mark Meyer
> Very droll. The trouble with a net is that most of it consists of
> empty space, and it's usually the empty space that replies.
Very droll right back atcha. :-)
--
Mark Meyer | mme...@dseg.ti.com |
Texas Instruments, Inc., Plano, TX +--------------------+
Every day, Jerry Junkins is grateful that I don't speak for TI.
HI! I'm a mutating signature virus. You can resist helping me spread!