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The Monty Hall Problem

This website is about the Monty Hall Problem, a famous brain teaser. The best-known version was asked to columnist Marilyn vos Savant in 1990:

The answer is yes: switching to the other door doubles your chances of getting the car. Many people find this result counterintuitive, and think that switching shouldn't make a difference. Even mathematicians can get tripped up by this problem (vos Savant answered correctly and got a lot of angry letters disagreeing!) There are many good explanations for the original Monty Hall Problem online: I like this one from Numberphile.

The problem is named for Let's Make a Deal host Monty Hall, though the game described in the problem isn't exactly how the show was played.

However, the wording of the puzzle doesn't always emphasize a key aspect: the host knows where the car is, and will always open a door with a goat behind it.

If the host doesn't know where the car is, and opens one of the other doors at random, it is not better to switch. Even if he reveals a goat!

A lot of people who mostly understand the original Monty Hall Problem get this wrong. And because the Monty Hall Problem is so famously counterintuitive, it's easy to make a mistake on this modified version and then double down, insisting to yourself your answer only seems wrong because the puzzle is so tricky.

But it's true: If Monty randomly opens a door with a goat behind it, there's no reason to switch. If Monty deliberately opens a door with a goat behind it, you should switch. It makes a difference whether Monty knows.

Why is this true?

Understanding why is a lot easier once you've demonstrated to yourself that it is true. So if you have doubts, on the main page you can play the game both ways. (Keep in mind that if Monty doesn't know where the car is, he might accidentally reveal it, ending the game early!) Try or simulate it as many times as you like: when Monty doesn't know where the car is, then if he reveals a goat, there's no advantage to switching.

There are also ways to prove this rigorously, by drawing tree diagrams or writing probability formulas that cover all cases. This website is not about that. I am not here to prove this result, and I'm going to use math terms as little possible. This website is trying to explain conceptually, intuitively, how this can be possible.

Because I don't blame anyone for getting confused! Why does it matter if Monty knows where the goats are, if he reveals a goat either way? Math doesn't care about "beliefs" or "intentions," right? If Monty does the same thing in both situations, how can it make a difference why he does it?

Here are some ways to think about it:

Monty's behavior is forced

One cause of confusion is that the problem is usually worded as "the host knows what's behind the doors." But to be fair, we don't exactly care about what Monty "knows," in the sense of his psychological state. The phrase "the host, who knows what's behind the doors, opens a door which has a goat" really means this:

The host will never open the door with the car behind it.

The Monty Hall Problem works no matter why that is. Maybe the host is being fed instructions through an earpiece. Maybe he's been cursed, and his hand is magically compelled to never go near the door with the car. Maybe he has really, really accurate intuition. The point is, the puzzle relies on the host's predictable behavior, not his knowledge. After all, the host in this simulated version is just a few lines of code - I'm not sure it "knows" anything!

Why does the host's behavior matter? Set the switch to "Monty Doesn't Know" and play or simulate a lot of tries. No matter your strategy, about 1/3 of the time, he reveals a goat and then you win, and 1/3 of the time, he reveals a goat and then you lose. But also, 1/3 of the time, Monty accidentally reveals the car, ending the game. So with either strategy, you have an even chance of winning in the cases where Monty reveals a goat - but there are also these extra cases where Monty reveals the car.

(You might be bothered by me saying Monty revealing the car "ends the game early": where did I get that rule? But since the puzzle asks about what happens when Monty reveals a goat, it doesn't actually matter what happens if he reveals a car. We could say that means you automatically lose, in which case you have a 1/3 chance of victory no matter your strategy. Or it could count as a win, and you have a 2/3 chance of victory. But regardless, if Monty reveals a goat, there's no difference in your success rate if you switch or stay.)

If Monty choosing randomly reveals the car 1/3 of the time: what if we make it so he can't reveal the car? What if (because he knows where the car is, or through divine intervention, or whatever) he will never open the door with the car behind it?

Well, when Monty's choosing randomly, he can only reveal the car if the original door you picked has a goat behind it. That's when he's faced with two doors, one with a car and one with a goat. Half the time, he'll reveal a car, ending the game. The other half of the time, he'll reveal a goat - leaving the car behind the other door, meaning it's better to switch. If instead, he always reveals the goat, that means that every time you initially choose a goat, he has to reveal a goat and leave the car behind the other door. So all of those cases where he would have revealed the car are now situations where it's better to switch!

If you're drawing a tree diagram, you could think of it this way: if Monty doesn't know, in 1/3 of branches you should switch, in 1/3 of branches you should stay, and in 1/3 of branches he reveals the car. If he does know, then the branches where he reveals the car turn into branches where it's better to switch, yielding the result in the original problem: switching wins 2/3 of the time.

It's about information

The above explanation... kind of makes sense? If you're mathematically inclined and love drawing tree diagrams, it might be enough. But I find it unsatisfying - it doesn't tackle the weirdness of this problem. Because it really feels like both situations are the same! You point to a door, Monty reveals a goat, you can switch or stay: why should we care about hypothetical branches on trees that didn't actually happen? What do I even mean, branches turn into different branches? We know what branch we're on: the problem says he revealed a goat. All our reasoning is conditional on that happening. So why do we care what could have happened, but didn't?

On the left: what it looks like when Monty reveals a goat deliberately. On the right: what it looks like when Monty reveals a goat accidentally.

Here's a different way to think about it: you're looking for information. You're like a detective, hunting for clues. And the same object can be a different clue depending on where it came from. The window was open on the night of the robbery - it matters if someone leaves it open every night, or if it was just open this once. The same open window means something different depending on how rare it is - it gives you different information.

What information is Monty giving us? It might make more sense to think in this way: when he reveals a goat every time, what information is he hiding? Try to picture exactly what it's like when Monty chooses randomly:

You've pointed at one of the doors. You don't know which door has the car. Monty randomly opens a different door. There's a goat behind it.

...Don't you feel better now? You know that at least you aren't pointing at that goat! You breathe a sigh of relief, because it feels like the probability you're pointing at the car has gone up!

And it has! A lot of people hear slightly garbled explanations of the original Monty Hall problem, which will say things like: pointing to a door at the start "locks in" a 1/3 probability that the car's behind it. There's a 2/3 probability it's behind one of the other two doors - so when Monty reveals a goat behind one of them, there's a 2/3 chance it's behind the remaining door.

But probabilities don't inherently "lock in" like that! They absolutely change in response to new information. If you're playing a card game, and you hope your opponent isn't holding an ace, you feel better if you see three aces dealt randomly onto the table. Even though the aces were dealt after their hand, they still give useful information: there's only one ace left, making the odds low that it happens to be in their hand.

When Monty doesn't know, the puzzle is kind of like a different game show: Deal or No Deal. That has a set of briefcases containing a range of different amounts of money, between $0.01 and $1,000,000. The contestant chooses one, and as the show goes on, randomly opens others onstage to reveal what's inside. When low dollar amounts are revealed, contestants sigh with relief! Each case contains a different number of dollars, and every time you reveal a low number, it means that can't be the number in your case. Your expected value goes up.

Similarly, when Monty randomly opens a door to reveal a goat, it means you can't be pointing at that goat, and that makes it likelier that you're pointing at the car. To be exact, the information raises your probability from 1/3 to 1/2!

But when Monty is deliberately only revealing goats, he's hiding that information from you. You don't learn anything when he reveals a goat... because you already knew he would reveal a goat!

Like the window left open on the night of the robbery, the same event means something different if it was guaranteed or uncertain. If Monty doesn't know where the car is, then he was guaranteed to reveal a goat if you initially chose the door with the car, but if you chose one of the goats initially, he only had a 1/2 chance of revealing a goat. That's why when he reveals a goat, it lets you know it's more likely you chose at the car.

But if he's guaranteed to reveal a goat no matter what you chose, revealing a goat doesn't tell you anything. There's no new information, and no sigh of relief. It's like he went through the deck and carefully pulled three aces: who cares? Whether there's three aces or four remaining in the deck, anyone can deliberately pull three aces.

This is what, in a sense, "locks in" the 1/3 probability. It was 1/3 when you pointed at the door, and Monty hasn't done anything to change that. The key to the original problem is seeing how we can use Monty's tricks against him. The fact that he hasn't given you more information about your own door teaches you something about the other one: by changing the number of doors available, without changing the probability that your own door is correct, he's increased the probability for the other door.

Let's talk about a common way people make the original Monty Hall Problem make sense: extending it to an extreme case. Suppose there are a hundred doors. You point to door #23. Slowly, deliberately, Monty opens every door except for #23 and #57, revealing a goat behind each one.

This raises an obvious question: why is Monty avoiding door #57? That's very suspicious! There's only a 1% chance that you were correct when you chose #23 at the start. So it's likely that he's avoiding #57 because opening it would reveal the car. You should switch!

But if Monty doesn't know where the car is, he can't be deliberately avoiding it. If he randomly opens 98 doors and doesn't find the car... sure, that's surprising, but it's not suspicious. (Well, you might find it suspicious, but all versions of this puzzle assume you can trust what Monty says: we know whether he is picking randomly or deliberately.) In fact, opening all those doors without revealing the car makes you feel a lot better about your initial choice! A very plausible reason he hasn't found the car after so many random doors is... it's been behind your door the whole time! Each door that opens gives you information, which has steadily increased your probability from 1/100 right up to 1/2.

You're in the same situation as a contestant on Deal or No Deal who eliminates all but one onstage briefcase, and the remaining dollar amounts are $0.01 and $1,000,000. This is an unlikely occurrence, sure - but now that it's happened, it makes for great television, because you have an equal chance to be holding one cent or a million dollars. Each time you eliminated a briefcase that didn't contain the million dollar prize, you increased the probability it was in each of the remaining briefcases onstage, as well as the one in your hands. Now there's only one onstage: there's a 50/50 chance the million dollars are in there, and a 50/50 chance it's in yours.

No, really, it's all about information

Let's get really conceptual. To start, I'm going to say something silly:

When you first pick a door, the probability a car is behind it is not 1/3.

Why? Because the door you're pointing to either has a car behind it, or it doesn't. If it does, there's a 100% chance you're pointing at the car. If it doesn't, a 0% chance. So obviously, it must be one of those values - "1/3" doesn't come into it! Sure, you don't know which of those it is, but one of them is true (and in the original version of the problem, it's not even like no one knows; Monty does!)

Hopefully, you're now saying "but... probability doesn't work like that!" But why not? You're either pointing at a car, or pointing at a goat, and there is a real, material difference between those two. The only reason that you don't take that difference into account when calculating probability is that you don't know which one you've chosen.

This is a basic fact about probability, that a lot of us learn early on but come to forget:

Probability does not describe reality. It describes our knowledge of reality.

Outside of the (fascinating, bizarre) world of quantum mechanics, we use probability is not because it reflects how the world actually works: we use it when we lack complete information as to how the world works.

And I think a lot of people nominally know this, but most of the time, we don't really think about it. Because most probability questions are about situations where no one could plausibly get more information. We assume that when a deck of cards is shuffled, no one could keep track of the location of every card. The skittering of a ball around a roulette wheel is too chaotic for anyone to track. The best prediction anyone could make is that there's an equal chance of every outcome; even though, of course, there is actually only one outcome that will happen.

If there's one fundamental reason the Monty Hall Problem causes so much confusion, it's that it isn't that kind of situation. Monty knows and you do not. Or, at least, Monty will act based on information you did not have. And information is so central to the idea of probability that what information Monty uses affects the probability, even if his actions are the same. Revealing a goat is the same action, but it comes from a different mechanism, a different set of rules and interactions that produce a result.

In fact, I might put it this way:

Probability is a description of a mechanism.

Probability is a way of describing how things work, not how things are. It does not say what's behind your door! You will never open a door to find 1/3 of a car. Probability is a way of describing the process by which something came to be behind your door.

When you say, "There's a 1/3 chance a car is behind that door," you're saying: "I know there are three doors, and that someone got one car and placed it behind one of those doors, but I don't have any reason to think one of the doors is more or less likely than any other. I do know, or at least am assuming as true, that it is behind a door, not off in the parking lot; that there's only one car; that it isn't split somehow behind multiple doors; that the people backstage aren't moving the car around in response to my guess," and so on. Probability is not about what's actually true, not about where the car actually is; it's about what you know or are assuming about how a situation came to be, and what that tells you about what might happen next.

And that, to me, is the most fundamental reason it matters whether Monty knows. Whether Monty is choosing randomly or choosing a goat every time is part of the mechanism of the game, and the whole point of probability is to describe the mechanism. It's not a description of what has happened, or even what will happen, but of the process by which things happen.

Here's one more way to think about it, with another silly statement:

The best strategy in the original Monty Hall Problem is not to switch every time.

Why? Because the best strategy is to just open the door with the car! You shouldn't switch every time: you should switch if you aren't pointing at the car, and stay if you are pointing at the car. That's way better than a 2/3 chance of victory, that's a 100% chance!

This isn't impossible. That is, these actions aren't impossible: nothing physically prevents you from always choosing the car door. It's just hard to think of this as a "strategy," but only because of your lack of knowledge.

But it's worth keeping in mind this "better strategy" as you think about what probability means. Suppose you're playing the original Monty Hall Problem: you approach the three doors, and you happen to point to the car. Monty deliberately opens another door to reveal a goat, and you smugly think to yourself, "Aha, this is the classic Monty Hall Problem, and because I understand how the math works, I know that I should switch." But you're wrong. You shouldn't switch, not this time. "Always switch" is not the best strategy, it's just the best strategy you can do. It's only because we don't know where the car is that we reason in terms of probabilities. But either you should switch or you shouldn't, and in the real world, you shouldn't switch a full 1/3 of times: you just don't know when those times are.

As silly as all this may sound, this is what helped me really understand what's going on here. It feels weird that the exact same actions could lead to a 1/3 probability in one situation and a 1/2 probability in another. It feels like it breaks the laws of physics somehow: the same actions are supposed to lead to the same state, right? But probabilities are not physical realities. There is not 1/3 of a car behind that door or 1/2: there either is a car or there isn't. With any given attempt at the game, you're either pointing at the car or you're not. Probability is just a way of describing what you know about the situation, all of the possible information you could use as clues to what you might be pointing at. And if one of the things you know is that Monty knows, it changes the probability.