Monty Hall Problem
Get three playing cards - an ace and two jokers.
Note where the ace is, and put all three cards in front of you face down, and ask a friend to try and guess where the ace is. Have them pick one of the cards, but don't let them look at it. Then turn over one of the jokers and tell them they can pick again, if they'd like. I.e., they can either keep the card they originally picked, or they can pick the other card.
So should your friend switch, or stick with their original pick?
Consider that your friend has two cards in front of them - one of them is an ace and the other is a joker. It's a 50/50 chance now, right? So switching won't make much of a difference.
Wrong. It turns out that switching is always the right thing to do. Switching gives them a 2/3 chance of winning instead of the original 1/3 chance or the perceived 1/2 chance.
You can go ahead and test it out yourself. Just play the game with a friend - like 10 times, and soon you will convince yourself that switching tends to win.
Another way to think about the problem: what if you had 8,000 cards in front of you? One of them is an ace, and all the rest are jokers. You pick one that you think is the ace, (a 1 in 8,000 chance) and then 7,998 cards are turned over and shown to be jokers. You've just got your first, 1 in 8,000 pick, and the remaining card. One of them is the ace. Looking at it this way, it should be clear that you should switch.
Still not convinced? Try it out:
Read more about the Monty Hall problem at Wikipedia.