Here is an example. Suppose for some reason you believe that the guy offering you the challenge actually does so by, in advance, doing the following. He chooses completely randomly an amount x from the following list: 1, 2, 4, 8, 16, 32, ... , 2 to the power 63 dollars. (Remember the grains of rice on a chessboard?) He puts x in one envelope and 2x in the other. You pick one envelope at random. Now if you looked in that envelope and saw a=1 you know you should switch, if you saw a=2 to te power 64 you'ld hold on to it; but if you saw anything else, it would indeed be equally likely that b is half or twice what you saw. So if you really were interested in maximizing the expectation value of what you get, you would switch in all those cases. In other words, there's only a 1 in 63 chance that you wouldn't want to switch. And anyway, who's interested in get 2 dollars instead of 1 when we are talking such mega amounts of money? So the rational guy who uses expectation value to decide whether or not to switch, will probably not bother with looking in the envelope (perhaps he has to pay to get a peek, first!) but just switch anyway ... which is obviously stupid!  Put this little example on your computer and do some explicit calculations. Do you think a casino would ever offer this game? What would be the entrance fee to be allowed to play (ie pick an envelope?) How much extra would you pay in order to take peep in envelope A before deciding whether or not to switch?