On a table before you lie two envelopes, identical in appearance. You are told that one envelope contains twice as much money as the other, but given no indication which has more. You are then allowed to choose one of the envelopes and keep the money it contains.
You then pick one of the envelopes at random, but before you look inside you are offered the chance to exchange your envelope for the other. Should you exchange envelopes, (assuming that the more money you get, the better)?
One line of thinking is this: If the envelope you initially choose has dollars in it, then there is a % chance that the other envelope has dollars in it and a % chance that it has dollars in it. Thus the expected amount of money you would end up with if you chose to exchange envelopes would be
You thus decide to exchange envelopes. But before doing so, you think to yourself, "But I could go through the same thought process with the other envelope, concluding that I should exchange my new envelope for my old one, and then go through the same thought process again, and again, ad infinitum. Now I have no idea what to do."
How can you resolve this paradox?
Edit: As mentioned in my comment below, the real "puzzle" here is to identify the flaw in the reasoning I outlined above. Or is there a flaw?