You Can Understand the Monty Hall Problem

Most explanations just aren't very good.

The Problem

You are on a game show trying to win a car.

Monty Hall, the host, presents you with three doors of which you can pick one. One of the doors is hiding a car you want, and the other two are hiding goats (which, for the purposes of this explanation, I will assume you don't want). You have no idea which door is hiding what, so you pick a door at random.

Let's say you pick door #1. Before Monty lets you see what's behind it he opens, say, door #3, always revealing a goat (if he were to reveal the car, the show would end). He then asks you if you would like to switch your choice to door #2 or to stick with door #1. Is it a good idea to switch doors? Does it even matter either way?

Please take a few moments to think about it.

The Answer

You might already know that the answer is yes; it does matter and you should switch doors. Under the standard rules of the Monty Hall problem—where the host knows where the car is, always opens a door with a goat, and always offers you the chance to switch—switching gives you better odds. I will show you why this is true, and you may be surprised by how intuitive the problem becomes once visualized properly.

Visualizing Worlds

It's time for emojis.

Believe it or not Monty has already given you enough information to make a non-random choice that will maximize your chances of winning the car. One thing you already know is that there are only so many possibilities for what is behind each door. You can use this information to think about the problem in terms of "worlds".

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There are three^[1] possible scenarios, or "worlds", that you could be in, as shown in the three groups above. In one world the car is behind door #1, in another it's behind door #2, and in yet another it's behind door #3. You don't know which of these worlds you're actually in, but these are the only possibilities.

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Let's say you pick door #1 again (denoted by the 👇 emoji). In one world you pick the car, but in the other two you pick a goat. In other words, there is a 1/3 chance that you will pick the car, and a 2/3 chance that you will pick a goat. Notice that you're more likely to pick a goat regardless of which door you initially pick.

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Monty (🕴️) then opens one of the other two doors, always revealing a goat. Since Monty's door is now eliminated from the possibilities, let's remove it from the picture to see what we have left.

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Much better. As you can see above, in two of the possible worlds the car is behind the door neither of you picked. This means that if you switch your door you will win the car 2/3 of the time, and if you stick with your original choice, you will only win the car 1/3 of the time.

So switching doors is the better choice!

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This actually happens no matter which door you initially pick. To demonstrate this, let's say you pick door #2 instead, as shown above. Monty then goes and reveals a goat. Let's see what we have left after removing Monty's door from the picture since it's no longer in play.

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In two of the three worlds, the remaining door that neither you nor Monty picked has the car. Meaning, if you switch doors you will win the car most of the time, so switching doors will give you the best likelihood of winning the car!

Why Switching is Better

The key insight is understanding what Monty's action actually tells you.

When you first pick a door, you have a 1/3 chance of being right and a 2/3 chance of being wrong. These odds don't change when Monty opens a door, but something important happens to that 2/3 probability.

Since Monty always reveals a goat, he's giving you non-random information. He's strategically eliminating one wrong option while preserving the car's location. This means the 2/3 probability that the car was in one of the other two doors gets concentrated entirely onto the remaining door you didn't pick.

So you're not choosing between two doors with equal odds. You're choosing between your original door (1/3 chance) and what amounts to both of the other doors combined (2/3 chance), with the goat door conveniently eliminated for you.