Addition to my last post:
(1) The original game: envelope A and B, you know one has double the amount of the other, but you don't know which. You open A and find $100. Should you switch to B, which may have either $50 or $200?
(2) A variation: everything is the same, up to the point where you are pondering whether to switch to envelope B, when the millionaire walks in, and hidden from view, flips a coin to decide whether to replace whatever was originally in envelope B with either double or half the sum in envelope A, i.e. either $50 or $200.
Say one envelope contains $x and the other $2x. If you keep the first envelope in game (2), and if you keep the first one OR switch in game (1), you should expect to win $1.5x. If you switch in game (2), you should expect to win 0.25*($0.5x + $2x + $x +$4x) = $1.875x.
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