Bigger or Smaller
12 Sep 2008
Puzzle

Alice writes two distinct real numbers between 0 and 1 on two chits of paper and places them in two different envelopes. Bob selects one of the envelopes randomly to inspect it. He then has to declare whether the number he sees is the bigger or smaller of the two. Is there any way he can expect to be correct more than half the times Alice plays this game with him?

Source

Heard from Gagan Aggarwal in 2000--2002.

Solution

Let the number revealed to Bob be p. Then Bob should say "bigger" with probability p, "smaller" otherwise. If the other number is q, then the probability of winning is ½ + ½ |p - q|.

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