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Wallet Game Paradox

wallet game paradox

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.

PROCEED WITH CAUTION

Professor Saunders selects two students at random from his class. He propses 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, habitually 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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