A generous person has put some money in two envelopes and offered to give you the contents of one. They tell you that one of the envelopes contains twice as much as the other. They decided which would have the larger amount randomly. Furthermore, after you pick one and open it up, you may change your mind and choose to have the contents of the other instead.

You open one up and find some money inside, say x dollars. Now you wonder if it is in your best interest to switch. You go through the following train of thought.

Which envelope contains the larger amount is random, so the other one contains 2*x dollars with probability 1/2 and x/2 dollars also with probability 1/2. So if I switch, the amount of money I should expect to end up with is (1/2)*(2*x) + (1/2)*(x/2). This is 1.25 * x, which is more than x, so I should switch.

Do you see what's wrong here? It seems to always be better to switch, but the situation is entirely symmetric! Selecting one and switching is no different from selecting the other and not switching. What's going on here?

Try to think through an explanation for this strangeness. You can read my comments if you like, but this is a particularly good problem to think through on your own first.