Hello everyone. Today we’re going to discuss The Monty Hall Problem. But why? Why should anyone discuss The Monty Hall Problem? It has been discussed ad nauseam. There are so many articles and videos about it that are fantastic, including Kevin’s video about paradoxes on Vsauce 2. Check that one out. Does the world really need another discussion of The Monty Hall Problem? No. But maybe it does. You see, I believe that true understanding comes from listening to as many voices as possible, discussing something from as many perspectives as possible, and so the more people you listen to talk about The Monty Hall Problem and why its solution is what it is, I believe through osmosis the greater your understanding will be.

Lately, for the last few weeks, I have been driving around on my commute to and from the Vsauce office talking out loud to myself about The Monty Hall Problem. If you see me talking to myself in my car, I’m not on the phone. I’m literally explaining The Monty Hall Problem to myself, honing in on the best way to intuitively get it, and I feel like I have found an explanation that…well, that I like. I’m not saying it’s the best, but I wanted to share it with you in the spirit of understanding. So let’s dive in.

### What is The Monty Hall Problem?

First of all, what is The Monty Hall Problem? Well, it was named after game show host Monty Hall, who hosted “Let’s Make a Deal.” One of the challenges he would put the audience to involved three doors on the stage: door number one, door number two, and door number three. Now Monty would honestly tell the audience member that behind two of the three doors was a goat. A goat. Yeah. But behind one of the doors was a bunch of money. Let’s say a million dollars.

The audience member got to pick one of the three doors, and if they picked the door that had money behind it, they would get to keep the money. But if they picked a door that had a goat behind it, well, they’d get a goat and they’d have to take care of it and travel back home with it. Point is, you weren’t supposed to want the goat, but goats are awesome, so I, in my version, have replaced the goats with pieces of poop. Fewer people want poop. Some still will, but the point is you’re supposed to want the money. If you pick a door that has poop behind it, you have to, like the goat, take the poop home, and take care of it and feed it, all that normal poop stuff.

### The Decision-Making Process in The Monty Hall Problem

So here we go. Let’s say you pick door one. Before I open door one, I’m going to tell you this: it’s good that you picked door number one because there’s poop behind door number two. You see that? There’s some poop. Oh, you don’t want that. But now, before I give you what’s behind the door that you have chosen, I’m going to actually let you switch. You can switch to door number three or you can stick with your original choice. After you’ve done that, whether you’ve switched or not, you will have to take home whatever is behind the door that you have chosen. Would you like to switch or stick with your original choice?

Most people would stick with their choice. Why? Because it feels like a 50/50 chance. You know that there’s no money behind door number two, which means the money’s either behind door number one or door number three. So you have a 50/50 chance, right? Many people think, “Why not just stick with your first choice because what if your first choice was right and you switched? Well, then you would really hate yourself for not going with your gut. It would feel like you lost the money. Like you had it and then you lost it by switching. If it’s behind the door you didn’t pick, then it’s like you just picked wrong. You don’t feel like you’re losing something.” So psychologically, most people decide not to switch.

### The Mathematical Insight of The Monty Hall Problem

But mathematically, you should always switch. In fact, if you switch, you will win not half the time; it’s not actually 50/50. If you switch, you will win 2/3 of the time. And if you don’t switch, you’ll only win 1/3 of the time. So you have a 66.6% chance of winning if you switch. Why should that be? Why is it not 50/50? Well, it comes down to two extremely important facts about this problem.

#### The Initial Choice and Probability

The first fact is quite easy for us all to agree on: when you make your first choice, the probability that you have chosen the money is 1/3. Okay, that’s the chance that you’ll be right. The money is behind one of three doors. You have a one in three chance of picking the door with money behind it right off the bat. That’s pretty clear. But that also means 2/3 of the time you are choosing a door that does not have money behind it. More doors have poop behind them than have money behind them.

#### The Host’s Knowledge and Influence

The second and much more important fact is that the host knows which door the money is behind, and the host will never open a door that has money behind it. I believe most disagreements over the solution of The Monty Hall Problem come from confusion over the rules of the game. So let’s make it extremely clear. First, you pick a door. Next, the host opens one of the two doors you did not pick. And the door the host opens of those two will always be a door with the goat (or poop) behind it. You are then asked if you would like to stick with your original choice or switch to the door that you didn’t pick and the host also didn’t open.

This is why it feels like a 50/50. We know that the money is either behind the door you picked or the door that the host didn’t open that you didn’t pick. It’s behind one of them because it’s definitely not behind the one the host opened. The host only ever opens a door with a goat behind it. But surprisingly, to switch or not is not a 50/50 choice. Switching allows you to win 2 out of 3 times.

#### The Strategy and Its Implications

If you chose a door, say door number one, and then the host randomly opened one of the remaining doors, sometimes the host would open a door that had the money behind it, which would be terrible television because the money would be revealed and the host would say, “Okay, well you were wrong, would you like to switch to the one door that remains closed?” Of course, you wouldn’t. It wouldn’t matter. The game would be over.

Instead, the host always opens a door of the two you haven’t chosen that has poop behind it. So what does that mean? Well, it means that sometimes you choose the door that the money is behind. That’s awesome. And in that case, it doesn’t matter which door the host opens. The host could have a rule where they always open the leftmost door or maybe they just flip a coin and open one or two. Either way, the host can open either door.

But most of the time, you didn’t pick the door the money is behind. 2/3 of the time, you pick incorrectly. In that case, the host is forced to open one door. The host has no choice. The host cannot open door three because the money is behind that door. So, he must open the other door to reveal the poop. This is why you actually have more information than you might think. The host is always avoiding the door with the money behind it. So by looking at which door when you have the choice to switch is still closed, you actually know a little bit more about what might be behind that door.

#### The Critical Choice

In the case that you picked correctly the first time, you picked a door with the money, the host has decided to open one door and not the other simply because of random chance. But most of the time you’re choosing incorrectly, and the host has left a door closed. The door that the host left closed was left closed because there’s money behind it, and you should switch. To make this really, really clear, if I choose correctly, the door that remains closed has no money behind it. If I don’t choose correctly, the door that remains closed does. So if I’m wrong with my first choice, I should switch. If I’m correct with my first choice, I shouldn’t switch. It’s kind of like You VS You

Here’s the thing: you choose correctly much less often. If we were to run many, many trials, you choose correctly much less often than you choose incorrectly because you’re only picking the door with the money behind it 1/3 of the time. 2/3 of the time, most of the time, you will have chosen a door with poop (or a goat, or maybe it’s goat poop) behind it. So most of the time, you should switch because the door that remains closed is the money door.

### A Simple Analogy to Clarify

There’s an analogy that I think makes this even more clear. It involves a sack. This sack contains three marbles. Two are white, and one is black. Let’s say the goats are represented by the white marbles, and the million dollars