I've decided to post my thoughts here, so that I can get some input from others. If I'm wrong about something, I'd like to know about it.

**The problem**
For those who aren't familiar with the problem, here it is:

You have two indistinguishable envelopes in front of you. Both of them contain money, one of them twice as much as the other. You're allowed to choose one of the envelopes and keep whatever's in it. So you pick one at random, and then you hesitate and think "maybe I should pick the other one". Let's call the amount in the envelope that you picked first A. Then the other envelope contains either 2A or A/2. Both amounts are equally likely, so the expectation value of the amount in the other envelope is

.

Since this is more than the amount in the envelope we have, we should definitely switch.

This conclusion is of course absurd. The symmetry of the problem alone is enough to guarantee that it doesn't matter if we switch or not. A more formal way of showing it is to note that if the smaller amount is X, then

*both* expectation values are equal to

,

so it doesn't matter if we switch or not.

The problem is that we have two calculations that look correct, but only one of them can be. It's obvious that the second calculation is the correct one, but it's surprisingly difficult to understand exactly what's wrong with the first one.