Car Goat Problem

Six or seven years ago, I took a philosophy class in which the professor introduced me to the “Car Goat Problem” (also known as the “Monty Hall problem“). The problem is this:

Imagine you are on a game show and there are three doors in front of you. Behind one of the doors is a car and behind the other two doors are goats. The game show hosts asks you to choose one of the doors (the one you believe has the car). Then, the host opens one of the other doors, which has a goat. The host then gives you an opportunity to change your mind. You win whatever is behind the door your choose. Should you switch doors or should you stay with your initial door? Or does it not matter?

So, what do you think? My solution is below.

As soon as I heard this, I knew the answer, but it took a little longer to formulate a solid argument and convince the others.

Nearly everyone responded that it wouldn’t matter — it was a 50/50 chance. One door had a car behind it and the other had a goat, so it had to be 50/50… right?

The key part of this problem, however, is that after you choose your door, the host chooses a door that reveals a goat. You pick a door (let’s just assume door #1). The host must then open either door #2 or door #3. There are three possible car-goat combinations:

Car Goat Goat
Car Goat Goat: The car is behind door #1.

With Car Goat Goat, the host can open either door #2 or door #3, because both contain goats.

Goat Car Goat
Goat Car Goat: The car is behind door #2.

With Goat Car Goat, the host must open door #3, because only door #3 contains a goat.

Goat Goat Car
Goat Goat Car: The car is behind door #3.

With Goat Goat Car, the host must open door #2, because only door #2 contains a goat.

In this game, there are two situations:

1. You correctly pick the door with the car on your initial guess (1/3 chance). The host opens a door that reveals a goat. The other remaining door has a goat behind it. If you switch doors, you will choose a door that contains a goat. You lose.

2. You incorrectly pick a door with a goat on your initial guess (2/3 chance). The host opens a door that reveals a goat. The other remaining door has the car behind it. If you switch doors, you will choose the door that contains the car. You win.

Therefore, it is to your advantage to switch doors when the host gives you the opportunity — 2/3 of the time you will win a car. 🙂