logical deduction - The Monty Hall problem

I'm amazed that, although well known, this problem hasn't been posted yet.

Monty Hall

• You are on a game show.


• The host of the show shows you three doors.


• He tells you that behind one of them, is a new car, and behind the other two is nothing (or in some versions of the story, a goat).


• He asks you to pick a door.



• After you pick a door, the host opens one of the other doors (IMPORTANT NOTE: The host knows where the car is, and will never open a door that contains the car)


• Then with the two doors left he asks you:

Would you like to keep your door? Or switch?

Answer

You switch your door, because you have double the chance of getting the car

Explanation:

When you pick a door, there is a 1 in 3 chance that the door contains the car.

Let's examine that case of when you originally picked the car, and when you originally picked a blank separately.

Originally picked the car(1/3 chance)

In this case, the host can open either of the 2 doors, and reveal nothing. In this case it is beneficial to keep your original door.

Originally picked a blank door(2/3 chance)
In this case, the host isn't allowed to open the door with the car. He must open the blank door. Meaning that when you switch after picking a blank door, you're guaranteed to get the car!

From this we can easily see that you are twice as likely to get the car if you switch your door.