You are told there are two envelopes. One contains twice as much money as the other one. You pick one but are allowed to change your mind after picking it. (You are equally likely to pick the one with less money as the one with more money.)
To figure out how much on average the other envelope should contain, one might average x/2 and 2x because one is equally likely to pick one as picking the other. Since this comes out to 5x/4, one might always change his or her mind. But wouldn't this end up with one never making up his or her mind?
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When I first receive an envelope I just improved myself (on average) by 1.5X (assuming one envelope contains X and the other 2X). Trading it I still have 1.5X as I don't yet know what is in the envelope. If however I open the envelop and observe the money, say $10, and now I am allowed to decide about the trade, I would (on average) have a 50% chance of ending up with $5 and a 50% chance of ending up with $20. With $10 already in my hand, by trading I would end up with (.5*$5 + .5*$20) or $12.5 a $2.50 profit (on average). The act of looking in the envelope changes the problem. Had it been double or nothing then looking in the envelope would be quite revealing. The odd part is that after you look at the money you would always trade, regardless of the amount, unless you saw $0 or a negative$