The Two Envelopes Problem

You are presented with two envelopes, each of which contains a sum of money. You are told that one envelope contains an amount exactly double the amount in the other envelope. You will be permitted to keep the money in the envelope you choose.

Two EnvelopesYou choose an envelope at random, but before you open it, you are given the option to change your selection. You reason that if the amount of money in the first envelope is x, then if you switch you have an equal chance of receiving either 2x or .5x. You calculate your expected return by adding 2x and .5x to reach 2.5x, then dividing by 2 to reach 1.25x. This is more than you have now, so you decide to switch.

Is there anything wrong with your reasoning? What if you are given the option to change your selection again?

The line of reasoning that makes switching seem favorable leads to the paradoxical prospect of switching back and forth endlessly, because it is always favorable to switch. This is absurd, so the reasoning must be flawed in some way. This problem has led to much debate among mathematicians and philosophers.

One analysis of the flaw is that x stands for two different things in the equation. When referring to 2x, x refers to the smaller amount; when referring to .5x, x refers to the larger amount. This invalidates the equation.

A common-sense way of resolving the paradox is to observe that, unlike in the Monty Hall problem, there is no change in the situation before you are given the opportunity to switch. You had a 50-50 chance of choosing the larger amount, and you either did or you didn’t. Switching does not improve your odds.

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