Card doubling paradox 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 is:
$$0.5(0.5x)+0.5(2x)=1.25x$$
so it seems like it is better to swap. Can anyone explain this apparent contradiction?
This paradox has always interested me.  Something to think about is that there does not exist a uniform probability distribution over the positive real numbers (since they are infinite).  In arriving at your paradox, it seems you are assuming that any real number is equally likely, but this cannot be the case.
I know this question was asked a very long time ago, but I would like to share my response nonetheless, because it seems to me that there is an issue with the problem statement itself, and that clarifying this issue would resolve the paradox that is posed here.

The issue is with you stating that it "seems like it would be better to swap." What do you mean by better? 

Suppose that by better you mean that the goal of this game is to have the card with greater value at the end of each turn. Say you score 1 point each time this happens, and get 0 points otherwise. Then the expected value associated with the cards is 1 for the winning card and 0 for the losing card -- it has nothing to do with the specific values of the cards. In that case, the overall expected value is 0.5 * 0 + 0.5 * 1 = 0.5 for staying and 0.5 * 0 + 0.5 * 1 = 0.5 for swapping; i.e., the same value, as you'd expect. 

Now suppose, as I think is implied in the original problem statement, that by better you mean that the goal of the game is to have the card with greater value at the end of each turn, and that points are awarded to each player equal to the value of their card at the end of the turn, and the player with the most points after a set number of turns wins. 

In that case the expected values for staying and swapping are, in fact, as you say, and there is no paradox here. Suppose for the sake of argument that you drew the number 4 on every turn, and that there's a 50% chance that the second card is a 2 and a 50% chance that the second card is an 8. Then your expected value is 1 * 4 = 4 for staying and 0.5 * 2 + 0.5 * 8 = 5 for swapping, and 5/4 = 1.25. But this is exactly correct: over the course of, say, 2 turns, swapping twice might get you a 2 once and an 8 once (for a total of 10), while staying twice would get you a 4 twice (for a total of 8), and 10/8 = 1.25 as expected. The reason why swapping is better is that the 50% chance of doubling your number outweighs the 50% chance of halving your number. In the case of our example, 2 is only worth 2 points less than 4, while 8 is worth 4 more points than 4, which means that you could get an 8 once every 3 turns and a 2 the other 2 turns when swapping and have the same number of points as your opponent (who would have 4 for all 3 turns), while in actuality you should get an 8 once every 2 turns. 

So, there's your answer (I think -- I could be wrong). In your original problem statement, you say  


  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.


but this is under the assumption that the only "valuable" occurrence in the game is having  a card that has a higher value than the other (i.e., the first case I mentioned), whereas when you actually do your expected value calculation for swapping, you switch to the assumption that a card is just as valuable as the number it has on it (i.e., the second case I talked about). In other words, everything that you have said in your post is literally true, but given different assumptions, which is why a paradox seems to arise when there is none.