probability paradoxes

You are on a Game Show, and there are three boxes in front of you. Your host, Monty Hall, tells you that one of the boxes contains a valuable prize, and that the other two are empty. He explains the rules: you get to choose a box, then he will open one of the remaining boxes to show it is empty, and then you will be offered the chance to change your choice to the remaining closed box, or stick with your original choice. You choose a box. Monty, who knows where the prize is, opens one of the other two boxes, as promised, and shows you that it is empty. Should you change, or stick?

This problem causes massive debate whenever it is aired. Many people's intuition runs: "There are now two boxes, with an equal chance of holding the prize, so it makes no difference if I change or stick". However, the correct solution is "I had a one in three chance of getting the right box originally, so there is a two in three chance that the prize in in one of the other boxes. It's not in the one the host opened, so there is a two in three chance of it being in the other unopened box. I should change my choice and double my chances of winning."

Many people feel unhappy that the third box, which originally had a one in three chance of holding the prize, has suddenly changed to having a two in three chance, even though it has not been touched. In order improve our intuition of what is happening, let's change the number of boxes, to a million, only one of which holds the prize. Now choose a box. You have a one in a million chance of being right -- hardly any chance at all -- and a very much larger chance of being wrong. Now Monty opens 999,998 empty boxes, leaving one box remaining closed. It is now clear the chance that the prize is in your box is not 50-50. You really do have only a one in a million chance of having the prize, and there is only one other place it can be -- in the remaining closed box. So there is a 999,999 in a million chance of the prize being there. Obviously, you should change!

Ian Stewart , in one of his Royal Institution Christmas lectures, also did a demonstration of this, using a pile of cards containing pictures of goats and a car, which again clearly demonstrated why you should change: the split is between the prize being in the one box/card you originally chose, or in any of the other boxes/cards.

A small wrinkle: the host must explain the rules before you make your choice. Otherwise, you don't know if he offered the new option only because you correctly choose the prize the first time!

• Devlin. All the Math that's Fit to Print . Chapter 106
• Piattelli-Palmarini. Inevitable Illusions . Grand Finale
• The WWW Tackles The Monty Hall Problem

[Monty Hall is the host of the TV show "Let's Make A Deal", which show helped inpire the statement of the original problem.]

A does better than B in case 1, and also in case 2. Yet when the data from the two cases is aggregated, B does better than A.

• Simpson's paradox explained on Wikipedia, including real world examples, and a very nice vector-based illustration (here A is red, B is blue; "better" means "greater slope"; "aggregation" is vector addition).