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The Paradox of the wallet
Have you ever found yourself lying awake all night, worrying that
you can't empathise with your partner's insomnia?
Worry no more. With this paradox going round and round
in your head you can be assured of many a sleepless night yourself.
Professor Saunders selects two students at random
from his class. He propsed a simple game: Both students put their
wallets on the table. The money is to be counted and whoever has the
most money of the two has to give it to the other.
The two students are given a little time for consideration.
Mr Straus reasons that, if he loses, he will lose
the money that he has in his wallet, but, if he wins, he knows that
he will win more than that amount. What he stands to gain is greater
that what he stands to lose. As he reasons his chances of winning
must be 50%, he decides that he should play the game.
However, Ms Morris, the other student, uses the same
reasoning. She believes that her chances of winning are as good as
those of Mr Straus and that, if she loses, she only loses the amount
of money in her wallet, but that if she wins she wins more than she
has in her wallet.
How can the game be to the advantage of both Mr Straus
and Ms Morris?
It can't.
For the purpose of this paradox one must assume that
neither Mr Straus, nor Miss Morris, haditually carries more money
than the other. If one has no further information, it can be assumed
that the game is fair, ie neither has a greater chance of winning
than the other. However, this does not shed any light on the inaccuracy
of the players' reasoning.
This paradox was originated by Maurice Kraitchik in his book "Mathematical
Recreations." he describes the paradox with neckties instead
of wallets. Unfortunately he offers no explanation of what is wrong
with the players' reasoning. In Aha! Gotcha, Martin Gardner writes,
"We have been unable to make this clear in any simple manner.
Kraitchick is no help, and so far as we know, there is no other reference
on the game."
Sorry, but you were warned.
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