Here's one where the obvious answer that everyone gives first is wrong, though not everybody accepts the right answer because it seems so counterintuitive. I once got into a long discussion on this on another board, and eventually won over those who argued against me, but let's see how it goes here.

OK, you are shown three cups, one of which has a prize under it. You are asked to choose (and point out) one of the cups, but not lift it yet. After you have made your guess, the house lifts one of the cups that you didn't choose and shows you that it is empty.

You are then offered the opportunity to choose again, this time for keeps.

Should you stick with your original guess? Should you change to the other cup? Or does it not matter (i.e. your chances of winning are the same whether you change your guess or not)?

Enjoy, Marion