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| Title: | Mathematics at DEC |
|
| Moderator: | RUSURE::EDP |
|
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
1591.0. "the OTHER envelope please" by HANNAH::OSMAN (see HANNAH::IGLOO$:[OSMAN]ERIC.VT240) Mon Apr 06 1992 14:07
I present you with two closed envelopes, and I say, one has twice as much
money in it as the other. Please pick an envelope and you can have whatever
money is in it.
So you pick one, open it, and there's $100 in it. You say thank you and start
to walk away with your money.
But I say wait a minute ! The other envelope has a .5 probability of having
$200 in it, and a .5 probability of having $50 in it, so the the expected
value of switching to the other envelope is
.5 (200 + 50) = $125
So, you're better off taking the *other* envelope.
You think for a moment, then agree, and you take the other envelope instead.
o.k. Now we play again. This time, you pick an envelope, but before you
even open it, I say wait a minute ! There's some amount of money in there,
X, we don't know what X is yet, but we know that the expected value in the
*other* envelope is
.5 (2X + .5X) = 1.25X
So, sights unseen, you're better off taking the *other* envelope.
You think for a moment, once again agree, and you take the other envelope.
But before you even open *that* one, I say wait a minute ! We of course still
don't know what's in it, but there's some amount X, and the expected amount
in the *other* envelope (pointing back to the one you picked up in the first
place) is
.5 (2X + .5X) = 1.25X
so you really ought to switch, right ?
Can this be ?
| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 1591.1 | I'll take some of that! | SGOUTL::BELDIN_R | Pull us together, not apart | Mon Apr 06 1992 16:23 | 24 |
| Re: <<< Note 1591.0 by HANNAH::OSMAN "see HANNAH::IGLOO$:[OSMAN]ERIC.VT240" >>>
One envelope has X dollars and the other 2*X. Select one with
probability .5. Expectation (given X) is 1.5*X, but we don't
know what X is. Let u(x) be the probility that X=x. Then the
(unconditional) expectation is 1.5*E[X] = 1.5*Sum[x*u(x)] IFF X
has finite expectation. But, if X has finite expectation, then
you can calculate (or approximate) the probability that the
other envelope has $50 or $200. Application of Bayesian theory
will resolve the paradox. The paradox only exists if the
expectation of X is indeterminate. And that only happens in
theoretical cases. You can always find a reason to limit the
expectation. (For example, if I know your weekly salary, I can
be quite sure that you will not offer me 150% of your weekly
salary, just to bother me with this kind of puzzle).
By the way, the only logical conclusion is that both envelopes
have $0 in them, since you are offering either of them to me for
free. Definitely not a win-win proposition, but if you want to
do it, I'm game. :-)
Dick
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| 1591.2 | No no, the other *OTHER* envelope. | CADSYS::COOPER | Topher Cooper | Mon Apr 06 1992 17:07 | 14 |
| The problem here arises from an implicit assumption that the
distribution of the population from which the envelopes are sampled is
an infinitely wide uniform distribution -- which is not a "proper"
distribution. If you make the population distribution a proper
distribution (which is essentially the same thing as placing a
reasonable prior joint distribution on the contents of the envelopes in
a Bayesian approach) you will find, I think, that the paradox
disappears.
The Bayesian approach doesn't really fix the problem -- it simply makes
what was implicit explicit so that the root of the problem becomes
obvious.
Topher
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| 1591.3 | discrete solution | DESIR::BUCHANAN | | Tue Apr 07 1992 09:58 | 30 |
| 1591.4 | Best Solution | HOBBLE::GERTLER | | Wed Apr 29 1992 22:19 | 1 |
| Best solution: Take both envelopes!
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