Here is a nice little problem (via LJ-users avva and flaass). Suppose you are offered to play tic-tac-toe with a computer that uses the following algorithm:
1. If it has two in a row, it will play the third to win.
2. If you have two in a row, it will play the third to block them.
3. Otherwise, it will choose a place for the next move at random.
Would it make sense for you to play with such a machine, if you always make the first move, get $1 for each win, and pay $10 for each draw or loss?
Update: A solution can be found in the comments, and I am too lazy to delete it temporarily (and don't know if it is possible to screen/unscreen comments here). So if you want to think on your own, don't read the comments please.