You are given two envelopes containing money and are told that one contains twice as much as the other. You choose one at random, open it, and find $100. You are now given the options to either keep the money or switch to the other envelope and take what is inside. Since you chose at random, you figure that the other envelope with equal probabilities contains $50 or $200. If you switch and it contains $50, you lose $50 and if it contains $200, you gain $100. Since the average of -50 and 100 is 25, you ?gure that your expected gain is positive and that it makes sense to switch. Now you realize that you would reach the same conclusion regardless of the amount you found in the first envelope, so you did not even have to open it, just take it and immediately switch to the other. So, you might as well just pick the other envelope to start with. But then, by the same argument, you should switch to the first! This obviously does not make any sense. Where is the error?
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