October 04, 2004

A two envelopes puzzle

[Math_] Here is a version of the problem from Bertsekas and Tsitsiklis: "You are handed two envelopes, and you know that each contains a positive integer dollar amount and that the two amounts are different. You select at random one of the two envelopes and after looking at the amount inside, you may switch the envelopes if you wish. Is there a strategy that will increase above 1/2 the probability of ending up with the envelope with the larger amount?"

This is not to be confused with the related and much more popular two envelopes paradox. I first heard this problem in a different form where the two numbers did not have to be positive or integers.

I think it is instructive to look at the different variants of this problem where the two numbers come from: a finite interval, a half open interval, and a circular structure like hours or angles.

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