“A swami puts m dollars in one envelope and 2m dollars in another. You and your opponent each get one of the envelopes (at random). You open your envelope and find x dollars, and then the swami asks you if you want to trade envelopes. You reason that if you switch, you will get either x/2 or 2x dollars, each with probability 1/2. This makes the expected value of a switch equal to (1/2)(x/2) + (1/2)(2x)=5x/4, which is greater than the x dollars that you hold in your hand. So you offer to trade.
The paradox is that your opponent has done the same calculation. How can the trade be advantageous for both of you?”
Casella, G. and Berger, R. (2002). Statistical Inference. Duxbury Press.