Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value x) and after you have seen it, the dealer offers you an opportunity to swap without anyone having looked at the other card.
If you choose to swap, your expected value should be the same, as you still have a 50% chance of getting the higher card and 50% of getting the lower card.
However, the other card has a 50% chance of being 0.5x and a 50% chance of being 2x. If we keep the card, our expected value is x, while if we swap it, then our expected value if 05.*0.5x+0.5*2x=1.25x, so it seems like it is better to swap. Can anyone explain this apparent contradiction.