The Monty Hall Problem

Some statistical problems are more intuitive than others. The Monty Hall problem is fun because it seems to defy logic, at first glance. The problem is named after the host of Let’s Make a Deal, Monty Hall. And it is a great example of how probability updates.

There are 3 doors, with a different prize behind each door, and you may only open 1 door. 1 prize is something good, like a new car, and the other 2 are terrible prizes that you wouldn’t want. Choose a door.

What is the probability that you chose the door with the new car? 1 in 3

I know where the good prize is, and I want to help you out. So, before you open the door, I give you a hint. I open an incorrect door and ask you if you’d like to change your guess.

If you keep your initial guess the probability of success remains unchanged. However, if you change doors what does the probability become? Since there are 2 doors, most people incorrectly think the probability is 1/2 or 50%. However, each door started with a 33% probability, and when I took 1 door out of the equation 33% was added to the unclaimed door which remains. Therefore, you have 33% probability of success if you keep the door you initially chose and a 66% probability of success if you switch doors.

Go ahead and try a simulation of this game on the UCSD website!