The Monty hall problem became famous when Marilyn vos Savant published it in response to a reader’s question in her column in Parade magazine in 1990. It is as famous for the virulence with which Marilyn was mansplained as it is for the subtlety of the problem itself. Even the legendary Paul Erdos, one of the great names of 20th century mathematics was only finally convinced by a computer simulation.

This article will show how a causal framework can help elucidate the structure of the problem, which I will argue helps avoid the logical pitfalls that makes the correct answer so intuitively uncomfortable. In doing so, I will show the steps of a causal approach to problem-solving, and we’ll also solve the Monty Hall problem on the way.

The problem

You are a contestant on a game show. You’re shown three doors. Behind one of the doors is a car; behind the two others are goats. You like cars. You need a car, or you can sell it. You don’t like goats. You don’t need a goat. None of your friends need a goat. And they’re worth much less than cars.

You are asked to pick a door. The game show host (whose name is Monty Hall) then opens one of the other two doors to reveal a goat.

Monty then asks whether you want to stick to the door you’ve already chosen or to switch to the other. What is the best strategy?

The causal framework

Decisions and objectives are the poles of the axis on which causal models are built. We start with them. Our objective is to win the car, the variable that captures that outcome is “What’s behind the door that you finally choose, car or goat?”

Then we have two decisions:

1. Which door to choose first
2. Whether to switch or stick.

Our causal map starts out looking like this.

A causal map decomposes the relationship between decisions and objectives into causally related elements. We build backward from objectives and forward from decisions and reconcile the causal chains in between.

In this simple case, we start by observing that the outcome is a function of, is causally dependent on:

1. Which door we finally choose
2. Where the car is placed

The placement of the car is a primary node in this problem; it is not causally dependent on anything, so there are no arrows into it and we need go back no further.

On the other hand, our final choice of door is clearly a function of both our first choice and whether we switch or stick. But it is also a function of which door Monty opens.

There’s no basis to prefer any one door over any other to start with, so for the sake or argument, we may as well choose door 1. Then Monty can choose which of doors 2 and 3 he can reveal.

Monty’s move

Here’s the key insight the causal framework provides: Which door Monty chooses also depends on the placement of the car. This is the yellow arrow from car to Monty’s move on the map. This is obvious, but the causal framework reveals its significance. Information about the car propagates into Monty’s move.

The car can be behind any of the three doors, but Monty knows which one. If you have chosen the car — if the car is behind door 1 — then Monty can choose either door 2 or 3. If it isn’t, Monty has to be careful not to reveal the car. Thus at least some of the time, Monty is giving you information: the car is behind the door he has chosen not to open.

There is an intuitive argument that it doesn’t matter whether you stick or switch because the car is behind one of the two doors so the probability the car is behind either of them must be 50:50. But this argument only holds in the absence of any other information and, at least some of the time, you have information. Monty just gave it to you.

Stick or switch?

The causal framework does not solve the problem on its own, you still have to do the math. If the car is behind door 1 then sticking wins you a car and switching, well, baaaaad luck. If the car is behind one of the other two doors then sticking brings only caprine compensation, where as switching wins the car. You have a one in three chance of picking the car, so switching wins you a car 2 times out of 3 and sticking just 1. Switching doubles your chances of winning.

Takeaways

The Monty Hall problem is ingenious in three distinct ways: The sneaky way Monty slides you information; that he only does so in some cases, but that is still information to you because you don’t know which case you’re in; and that the winning strategy requires you to give up the car if you picked it in the first instance.

A causal map doesn’t give you the winning strategy on its own, but it’s a powerful and intuitive method of understanding how the problem works, and why seemingly intuitively obvious answers fail.