The Monty Hall Problem

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The "Monty Hall Problem" is a classic example of the non-intuitive nature of probability:

You are playing a game. In the game, there are three doors with prizes hidden behind them. One door hides a new car while the other two doors hide goats. If you pick the door with the car behind it, you win the car.

You pick a door. At this point the game host opens another of the doors, revealing a goat. Now the game host offers you the opportunity to change your pick to the other door.

Should you switch?

At first glance, most people believe that there is no advantage to changing their pick: of the two remaining doors, one of them has the car, so they have a 50% chance of winning. In fact, this is incorrect: while 1 of the doors will have the car behind it, 2/3 of the time it will be the other door. It is generally to your advantage to switch.

The reason lies in the fact that the game host knows more than you, and can only show you a door with a goat behind it. You start off by picking a door, with a 1/3 chance of picking correctly. Now the game host reveals a goat, so you are down to one car and one goat - switching at this point essentially reverses your initial odds. If you picked a goat to start with, switching will get you the car; if you picked a car, switching will get you a goat. Since you were only 1/3 likely to have picked correctly to start with, you will get a car 2/3 of the time if you switch.

Many people have trouble with this explanation, so I have illustrated all the possible cases, and the strategy required to win the car in that case.

Green cells indicate the first pick, red cells indicate the door opened by the game host, and the white cell indicates the door that you have the option to switch to.