Correct me if I'm wrong, but here's my analysis. There are two envelopes. One contains a value x, and the other has 2x. You pick an envelope at random. Let's c

Correct me if I'm wrong, but here's my analysis. There are two envelopes. One contains a value x, and the other has 2x. You pick an envelope at random. Let's call the envelope you choose z. The variable z is a random variable that has an equal chance of being x or 2x, so the expected value of z is 1.5x. If you switch, there are two possibilities. In one case, you picked x, so switching gains you .5x over the expected value of 1.5x. In the other case, you picked 2x, so switching loses you .5x from the expected value of 1.5x. So, the expected gain from switching should be zero.