Introduction

Have you ever known a puzzle that just wouldn’t “click”, no matter how hard you think about it? Welcome to the granddaddy of such puzzles – the “Monty Hall Problem”.

In this post, we explore what the puzzle is, why it is so confusing and how a key piece of insight could unlock it.

What Is It?

Originally posed as a probability question in a letter to the The American Statistician, a scientific journal, the problem entered popular culture via the American television game show Let’s Make a Deal.

The puzzle in the form in which it became popular goes like this

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

There are a few assumptions which are part of this definition of the puzzle

• Host opens a door not selected by the contestant
• Host opens a door to reveal a goat and never the car chosen originally and the remaining door
• Host offers a chance to switch between the door

Seems Easy. But, Is It Really?

The first event which occurs in the problem is the contestant picking a door. Let’s number this door as 1. So, whatever door is picked by the contestant is called “door 1”. This leaves two remaining doors.

The second event which occurs in the problem is the host opening a door with a goat among the two remaining doors. Let’s name the door that the host opens as door 3. This means that we have doors 1 and 2 still closed.

The third event which occurs is our choice to switch (or not) between the remaining doors. Since there are exactly two doors and we have no information about what’s behind either of them, it seems obvious that there’s a 50-50 chance of the car being behind any of them. So it shouldn’t matter if we switch or not.

Now here’s the kicker – we should switch to door 2 because the odds that the car is behind it is double that of the car being behind door 1. There is 2/3 probability of winning the car if you switch and 1/3 if you don’t.

How is this possible? It seems counter-intuitive and quite possibly wrong. If you disagree with this solution, don’t worry, you’re not alone.

You’re In Good Company

There’s a reason this puzzle has been in the zeitgeist for so long. It seems evident that we shouldn’t switch doors (or that it doesn’t matter if we do), but the solution says that we absolutely should. Not just us, this problem has stumped great mathematicians as well.

Paul Erdős, who is famously known for solving many unsolved mathematical problems from before his time, was known to initially disagree with the solution as well. When Andrew Vazsonyi, a researcher and educator discussed the problem with him, Paul thought that it shouldn’t matter if we switched the doors or not. He was only convinced after looking at the results of a Monte Carlo simulation of this problem and even then he was not able to intuitively understand the solution

The puzzle is so popular that even fictional characters have been written as being confused by this puzzle. Captain Holt, my favourite character from one of my favourite shows – Brooklyn 99, joins the long list of people who are confused by this puzzle.

What If There Were 100 Doors?

An idea which has been commonly used to get a better intuition for the puzzle is to expand the question to a 100 doors. The new formulation of the puzzle with 100 doors goes like this

Suppose you’re on a game show, and you’re given the choice of a hundred doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens 98 other doors, say No. 3, 4, 5, … 99, 100, all of which have a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

You can make the argument that the probability that you picked the right door with your initial choice is 1/100. Since there is a 99/100 chance that the car is behind one of the doors that you didn’t pick originally, you should switch because after the host has opened 98 doors, you’re moving to a door with higher probability of the car being behind it.

Does this make things clearer? I’m not really sure. Even with this new setup of a 100 doors, we could argue that at the end there are two choices for the door and that we should have a 50/50 chance of the car being behind either of them – which is the same argument we made in the case of 3 doors.

We’re back to square one, aren’t we?

Key Idea

Let’s try something different. I will try to reformulate the puzzle in a way which is equivalent to the original formulation but it is stated in such a way that it will make the solution obvious. Hopefully we can then make the same conclusion as the solution.

The new formulation of the puzzle with 100 doors goes like this

Suppose you’re on a game show, and you’re given the choice of a hundred doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1. The host renames your door as a group and calls it “Group 1”. The host then groups the rest of the 99 doors into what he calls “Group 2”. The host does NOT open any door but he says that you’ll get the car if it is behind ANY of the doors in the group that you pick.

He then says to you, “Do you want to pick Group 2?” Is it to your advantage to switch your choice?

Now, in this case it is quite obvious that you should switch because there’s a 99/100 probability that the car is behind one of the 99 doors in Group 2 and you would get the car just by picking the right group.

The big question is – why is this reformulation of the puzzle equivalent to the original formulation?

Let’s Dig In

The claim I’m making, which is implicit in the reformulation, is that the act of the host opening doors (with goats behind them) is irrelevant to the puzzle.

Now, how do we prove this claim? The question we need to answer is: “Does the act of host opening 98 doors provide us with any additional information or not?”. If the answer to this question is “No”, then the claim is proved.

Let’s break down the possibilities of where the car is, into the following two cases

• Case 1: Car is not behind Door 1
• Case 2: Car is behind Door 1

At the start of the game, we have no information about which of the above two cases is true. If after the host opens 98 doors, the contestant receives any additional information about which of the above cases could be true (or more true), then the act of the host opening the doors has some meaning, otherwise it is irrelevant. Keep in mind that the host needs to be able to select exactly 98 doors that he needs to open and show them to the contestant.

Case 1: Car is not behind Door 1

If the car is not behind door 1, then it is behind one of the doors 2, 3, .., 99, 100. Among doors 2-100, there are 98 doors with a goat behind them and there is 1 door with a car behind it (doesn’t matter which one). The host has exactly 98 doors available to him, out of which he will open all 98 of them

Case 2: Car is behind Door 1

If the car is behind door 1, then there is no car behind any of doors 2, 3, .., 99, 100. There are 99 doors with a goat behind them. So the host has 99 doors available to him, out of which he can open any 98 of them

Since the host has more than or equal to 98 doors in both the cases, the contestant cannot conclude if there originally were 98 (case 1) or 99 (case 2) doors with goats behind them. So they cannot say which of the two cases is actually true. So the host opening 98 doors has provided 0 additional information to the contestant about which of the two cases is true.

Let’s, for the sake of argument, assume that there are only 97 doors with a goat behind them in one of the cases and call it (hypothetical) case 3. So the host will not be able to open 98 doors for case 3. But if the host still opens 98 doors with goats behind them, then the contestant knows for sure that case 3 is not true and he has received additional information about one of the cases.

Based on this argument, it is proved that the act of opening doors is irrelevant. This means that the contestant can pick one of Group 1 or 2 (as in the reformulation of the puzzle). Since Group 2 has 99 doors, the contestant should pick this group.

This leads us to the conclusion that it is beneficial for the contestant to switch groups, and hence the doors.

Not Convinced Yet?

If you’re still not convinced by this argument and you’d like to see how this game would actually play out in real life, there are posts on the internet which describe how to perform a simulation and calculate the probabilities over a large number of experiments. You can find one such article here.

Summary

I know that this puzzle has been discussed to death – both on the internet and among mathematicians. But, I think that bringing a great number of diverse ideas from different people can only enrich the discussion. I hope I’ve been able to add something meaningful to the discussion about this puzzle with this post.

Let me know what you think about it.