Friday, November 12, 2010

Puzzle #2 - Statistical Guesswork

Alice secretly picks two different real numbers by an unknown process and puts them in two (abstract) envelopes. Bob chooses one of the two envelopes randomly (with a fair coin toss), and shows you the number in that envelope. You must now guess whether the number in the other, closed envelope is larger or smaller than the one you’ve seen.

Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?

While you may think that it is paradoxically defined or has no answer, it turns out that it’s a perfectly valid problem and the answer to it is “Yes.” In fact, there are at least two possible answers to this question that both satisfy the objective.

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