The Monty Hall Problem, Simplified

In this post I offer a visual approach to the famous Monty Hall Problem. I present it from what I believe is a much simpler perspective in order to sidestep common misconceptions and confusing terminology. My hope is that reading this will give you a robust understanding of the problem, and that you will understand with clarity why switching doors makes sense.

It's not as complicated as it seems, I promise!

First, 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 only pick one. One door is hiding the car you want, and the other two are hiding goats, which we 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 door #1, he opens door #3, revealing a goat. He then asks you if you would like to switch your choice to door #2, or to just stick with your original choice. Is it a good idea to switch doors? Does it matter either way?

I encourage you to take a moment to think. What would you do?

The Surprising Answer

As you might already know, the answer to those questions is yes; it does matter which door you pick here, and you should switch doors because that makes it more likely that you will open the door hiding the car.

This may seem strange, and you might even disagree. You might assume that you have no new information about what is more likely to be behind either of the closed doors, and so it shouldn't matter which one you choose. The odds should be the same.

Let's see why this isn't the case.

Visualizing Worlds

Yes, it's time for some emojis.

Let's rewind. Monty presents you with three doors. One is hiding a car, the other two goats, and at first you have no idea which door is hiding what. However, one thing you do know already is that there are only so many possibilities for what is behind each door. We can use this information to think about the problem in a different way.

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There are three¹ 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 possible 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. So, you're more likely to pick a goat. Notice that this is the case 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! Huh. 🤔

Why Switching is the Better Choice

There are two crucial pieces of information that people often miss here: 1.) As we saw earlier, you are more likely to pick a goat in your initial choice because of the simple fact that there are two goats and only one car, and 2.) the host always reveals a goat in the Monty Hall Problem.

For the first crucial piece of information, you more likely picking a goat means that the car is more likely to be behind one of the other doors. If Monty then goes and removes a goat from one of those doors, this makes it even more likely that the car is behind the other remaining door.

For the second crucial piece of information, Monty always revealing a goat is important because it means that the probabilities don't cash out to random chance. Monty is helping you out by revealing a goat. Combine this with the fact that you more likely picked a goat for your first choice, it then becomes clear that switching doors is the better choice.

Conclusion

I hope this blog post has helped you understand the Monty Hall Problem better. It's a fun problem that has a surprising answer, and it's a great example of how probabilities can be counterintuitive. If you're still not convinced, let's get in touch and talk about it! I'd love to hear your thoughts. 🫡

You may know there are combinatorics at play here with the goats, and so technically there are more than three possible worlds. I claim this is a distraction and it doesn't make a difference towards winning the car because the probabilities still cash out the same for the car. The important thing is that there are three possible doors behind which the car could be in. Goats are great, but the unique identities of the goats don't matter here. Sorry goats. 🐐 ↩