# The Monty Hall problem

This rather interesting problem arises from a game show, Monty Hall is the host..

There are three doors on stage, labeled A, B, and C. Behind one of them is a pile of money; behind the other two are goats. You get to choose one of the doors and keep whatever is behind it.

Let's suppose that you choose door A. Now, instead of showing you what's behind door A, Monty Hall slyly opens door B and reveals... a goat. He then offers you the option of switching to door C. Should you take it? (Assume, for the sake of argument, that you are indifferent to the charm of goats.)

Counterintuitively enough, the answer is that you should switch, since a switch increases your chance of winning from one-third to two-thirds. Why?

When you initially chose door A, there was a one-third chance you would win the money. Monty's crafty revelation that there's a goat behind door B gives no new information about what's behind the door you already chose -- you already know one of the other two doors has to conceal a goat -- so the likelihood that the money is behind door A remains one-third. Which means that, with door B eliminated, there is a two-thirds chance that the money is behind door C.

Still not convinced? Perhaps it will help if you look at the game from Monty's perspective. For him, the game is very simple. No matter what door the contestant picks initially, his job is to reveal a goat and ask the contestant if they want to switch.

Consider what happens when the contestant initially selects the door with the car. Monty has two doors to work with, both of which conceal goats, so he reveals one of the goats, and the door he leaves closed conceals the other goat. That should make sense: If you pick correctly with your initial door, which happens one-third of the time, switching to the other closed door ensures you'll wind up with a goat.

Now, consider what happens when the contestant picks incorrectly, and selects a door concealing a goat. Monty now has two doors to work with, only one of which conceals a goat, so he must open that door. The door he leaves closed therefore must conceal the money.

This is a key realization. If the contestant picks incorrectly, which occurs two-thirds of the time, the door Monty leaves has to conceal the money. So it makes good sense for the contestant to switch doors, for two-thirds of the time doing so will win them the money.