I got this problem from Rustan Leino, who got it from Raphael Reischuk, who also has a little puzzle collection.

I solved it and wrote up my solution.

You and a friend each have a fair coin. You can decide on a strategy and then play the following game, without any further communication with each other. You flip your coin and then write down a guess as to what your friend's coin will say. Meanwhile, your friend flips their coin and writes down a guess as to what your coin says. There's a third person involved: The third person collects your guesses and inspects your coins. If both you and your friend correctly guessed each other's coins, then your team (you and your friend) receive \$2 from the third person. But if either you or your friend (or both) gets the guess wrong, then your team has to pay \$1 to the third person. This procedure is repeated all day. Assuming your object is to win money, are you happy to be on your team or would you rather trade places with the third person?

You'd rather be on your team, because you can use the following strategy to win money in the long run.

Each of you, after flipping a coin, should make a guess that matches that flip's outcome. This way, you'll correctly guess each others' coins as long as the coins match. The probability of this happening is 1/2, so you'll win half the time. So, you'll win \$2 with probability 1/2 and lose \$1 with probability 1/2, and thus your expected value is to win \$0.50 each time you play.