The Monty Hall problem

The Monty Hall problem is one of my favorite problems whose solution is counter-intuitive.  Suppose you are being shown 3 closed boxes by Monty Hall, only one of which contains a prize and the other 2 are empty.  You get to choose a box, and Monty, who knows which box contains the prize, opens an empty box that you did not choose.  He then asks you if you want to change your mind.  Intuitively, one would think that it doesn't make a difference if you switch or not.  But it turns out your odds are doubled if you change your mind.  The reasoning is as follows.  If you have initially picked a box with the prize in it, then not changing your mind will result in you getting the prize.  You have a 1 in 3 chance of initially picking the winning box.  If you have initially picked an empty box, switching will result in you getting the prize, and you have a 2 in 3 chance of initially picking an empty box.  So the odd of winning by switching is double the odd of winning by not switching.