Choose a door

What's the problem?

Suppose you are a contestant on a game show and you have to pick one of three doors. Behind two doors are goats (or anything else you don't want to receive) and behind the the other door is a car (or something else you want). You are asked to pick a door and you get to keep whatever is behind that door.

 If you pick a door at random (call them door 1, door 2 and door 3) the laws of probability state you have a 1/3 chance of having picked the door with the car behind it.

Now suppose the game takes a turn. The game show host opens one of the two doors you didn't pick and shows you that there is a goat behind it. You are then given the option of changing your pick of doors.

Should you switch?

Purely mathematically, the laws of probability state that you should swtich your choice of door.

Simple intuition may tell you that the new probabilities of the car being behind each door are 1/2 however this does not take into account the fact that the game show host has information that you do not-the position of the goats.

In mathematics the probability of winning the car if you stick with your original choice of door stays 1/3.

To see why switching is better consider the case when you have picked door 1:

• If the car is behind door 1; the host will open either door 2 or door 3 and reveal a goat and if you switch you will receive a goat.
• If the car is behind door 2; the host will open door 3 and show you a goat. You switch to door 2 and you win the car.
• If the car is behind door 3; the host will open door 2 and show you a goat you will switch and you will win the car.

Each of these three outcomes has probability 1/3 and as you win the car in two of the cases the probability of winning the car is 2/3. As this probability is higher you should always switch your door to increase your odds of winning the car.