Math Exploration: Monty Hall Problem

The Monty Hall Problem is a probability puzzle. It goes as such: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

The way to determine whether to switch or not can be derived from listing out all the possibilities.

If you picked Goat #1: The host opens Goat #2 and switching would give you the Car

If you picked Goat #2: The host opens Goat #1 and switching would give you the Car
If you picked the Car: The host opens one goat and switching would give you the other goat.

It is easy to see now how switching gives you a higher chance of getting the Car. However, this problem is interesting because many people disagree with this solution and still think that there is no difference. Many people think that the chance of getting a car is still 1/3 as there are 3 doors or that after opening a Goat, there are only 2 doors therefore the chances of having a car is 1/2.