The Monty Hall Problem

By Sashrika Pandey

Photo credit: brilliant.org

By Sashrika Pandey

A classic problem in probability, and one that has prompted numerous discussions and the use of simulations, the Monty Hall problem encourages the use of critical thinking in determining whether new evidence can alter one’s course of judgement.

In the problem, a game show contestant is provided with three doors, two of which hide a goat and one which hides a car. The contestant chooses one door, but the host then opens one of the doors that hides a goat. Despite what might seem intuitive, it has been proven that deciding to switch doors - that is, changing one’s chosen door to the remaining closed door after the host has revealed one of the doors hiding a goat- yields a higher rate of success than sticking with the original door. Specifically, contestants who switched doors appeared to have a ⅔ rate of success while those that stayed with their original doors had a ⅓ rate of success. Many who encounter this problem find such a process counterintuitive, and yet the introduction of the host’s information is what gives this problem a unique twist.

Imagine that your initial choice, as the contestant, is correct and hides the car. That means that, out of the three doors, your choice was correct, or that you had a ⅓ chance of choosing the correct door.

Now, imagine that your intiial choice was incorrect. You have a ⅔ chance of picking the incorrect door, since two of the doors hide goats. After the host reveals a goat behind one of the other two doors, you may think that you have an equal chance of picking a car behind either of the remaining doors. However, that is not the case.

The host opened one of the doors with the knowledge that the car was not behind that door. In doing so, they revealed an important piece of information. We already know that there is a ⅓ chance that the door that you have already chosen hides the car. However, by switching doors, you increase these odds to ⅔. This is because you had a ⅔ chance of not finding the car in the beginning of the game. The two doors that you have not chosen have a cumulative ⅔ chance of hiding the car. This fact remains the same after the host opens one of the doors, with the key exception that you now know the contents of one of the doors. As www.montyhallproblem.com states, “You might say that the remaining door “inherits” the probability of the eliminated door — so now for that one door, the probability of it being the door which hides the prize is 2/3.”

The significance of this problem lies in the fact that switching doors appears counterintuitive when one does not consider the host’s conscious decision to open one of the doors. Probability does not always appear straightforward, but when one weighs all the possibilities, one can determine interesting conclusions. Probably.

Sources:
“The Monty Hall Problem.” The Monty Hall Problem. Date unknown. Monty Hall Problem
“Understanding the Monty Hall Problem.” Better Explained. Date unknown. Understanding the Monty Hall Problem - Better Explained
Weisstein, Eric W. "Monty Hall Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MontyHallProblem.html