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Suppose you are presented with a game. You are given a red and a blue envelope with some money in each. You are allowed to ask an independent party to open both envelopes, and tell you the ratio of blue:red amounts (but not the actual amounts). If you do, the game master replaces the envelopes, and the amounts inside are chosen by him using the same algorithm as before.
You ask the independent observer to check the amounts a million times, and find that half the time the ratio is 2 (blue has twice as much as red), and half the time it's 0.5 (red has twice as much as blue). At this point, the game master discloses that in fact, the way he chooses the amounts mathematically guarantees that these probabilities hold.
Which envelope should you pick to maximize your expected wealth?
It may seem surprising, but with this set-up, the game master can choose to make either red or blue have a higher expected amount of money in it, or make the two the same. Asking the independent party as described above will not help you establish which is which. This is the surprising part and is, in my opinion, the crux of the two envelopes problem.
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