# Monty Hall Problem - Explained

December 14, 2019

You’re in a game show, about to face the final challenge. The host, Mr Monty Hall, gestures to three doors. Behind one, he says, is a luxury model car. Behind the other two are goats. He asks you to pick a door, and you choose one. Before telling you whether you got a car or a goat, however, he opens a door you didn’t pick, revealing a goat. He then asks if you would like to stick with your original door or switch to the other unopened door. So, assuming you want the car, should you stick or switch? This is The Monty Hall problem.

Common Errors

Immediately, you’re probably thinking, “It’s a fifty-fifty chance! Might as well stick with the original.” This is the most common logical error when dealing with this problem. In fact, the probability is actually 0/3 for the opened door, 1/3 for the door you picked, and 2/3 for the other door. Why? Well, there are two ways of thinking about it.

So yes, you are better off switching doors than sticking with your original choice. One way of understanding it is to think about what is behind the doors. If there is a goat behind the door you picked, then switching will give you the car. If there is a car behind the door you picked, then switching will give you a goat. Now, there is a 2/3 chance that you picked a goat, and therefore will get a car from switching. Another way of thinking about it is to imagine there are a hundred doors, with 99 goats and 1 car. You pick your door, and Monty opens 98 other doors to reveal goats, leaving, say, door 47 unopened. What could possibly be so special about door 47? Maybe because there is a car behind it.

And that’s it, The Monty Hall Problem! Did you get it right?

Video Solution by previous Spires Scoop author:

Credit goes to Youtube user : Cowman