# The Monty Hall Problem

*I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.*

## The Monty Hall Problem

The Monty Hall Problem is named after the host of the US TV show 'Let's Make a Deal' and is a fantastic example of how our intuition can often be wildly wrong when trying to calculate probability. In this article, we are going to look at what the problem is and the mathematics behind the correct solution.

Suppose you are the winning contestant on a quiz show and for your grand prize you are given the choice of three doors. Behind one of the doors is a brand new car, while behind the other two are goats. You win whichever prize is behind your chosen door.

You choose a door, but the TV host asks you to wait for a moment. He then opens another door to reveal a goat and gives you the option of switching doors. Should you switch?

## Should You Switch Doors?

Intuition seems to suggest that it shouldn't matter whether you switch doors or not. There are two doors left; one has a car behind it, and the other has a goat, so you would think that it is a 50/50 choice either way. However, that isn't the case.

If you switch doors, you are actually twice as likely to win as if you didn't switch. This is so counter-intuitive that even many university professors of maths argued passionately against it when first faced with this problem.

Let's look at how it works.

## Why Should We Switch Doors?

Look back at the picture above. Suppose you pick door 2. The TV host then opens a door to reveal a goat. He knows where the goats are, so the open door will always be a goat; he won't reveal the car by accident.

This leaves two doors and we know that one has a car behind it and the other one has the other goat behind it. Therefore if we switch doors, we are guaranteed to switch prizes, either from car to goat or from goat to car.

You choose to switch doors. For the new door to have the car behind it, you need to have started off pointing at a goat door. If we can work out the probability of originally pointing at a goat, we therefore have the probability of the new door having a car behind it.

## The Probability of Starting on a Goat

As there were three doors to choose from at the beginning and two of those doors had goats behind them, the probability of picking a goat with your first choice of door is 2/3.

This is the outcome that would lead to switching doors giving you the car, hence if you switch doors, the probability of winning the car is 2/3, twice as big as the probability of winning if you stick with your original choice (1/3). Difficult to believe, but true!

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## Why Does This Work?

The thing to remember here is that even though you have ended up with only two closed doors, the host's choice of which door to open to reveal a goat was dependent upon your original choice of door; it is the probabilities of the original three doors that are important.

## An Alternative Way of Thinking About It

In case you are still not convinced, here is another way to look at the Monty Hall Problem.

There are three possible combinations behind the doors. Either the car is behind door 3, door 2, or door 1 and the goats fill up the remaining two places in each example.

## Examples

In the picture above, we are looking at what could happen if your original choice was door 1 (signified by the black arrow). In the top row of the picture, you choose door 1, the host opens door 2 to reveal the other goat and so switching will take you to door 3 and the car.

In the second row, we have a similar example. You start on door 1, the host opens door 3 to reveal the other goat and you switch to door 2, again winning the car.

In the bottom row, however, you start off pointing at the car, the host then opens one of the two remaining doors and switching will take you to the other goat.

So if you start on door 1, there are three possible outcomes when switching, two of which lead to winning the car, hence the probability of switching giving you the car is 2/3.

It can be quickly seen that the same would happen if you originally chose doors 2 or 3, so giving you an overall probability of winning by switching of 2/3.

**© 2019 David**

## Comments

**David (author)** from West Midlands, England on February 15, 2020:

The door that was opened is dependent upon your original choice, so the probabilities of sticking or swapping have been affected by that original choice. The probability of winning by swapping is 2/3, however counter-intuitive that seems.

Try it with a physical experiment. Get a friend to create a similar set-up; you could use toy cars and animals under cups for example. Make sure the friend knows where the car is so that they can act out the presenter role. If you keep on doing this experiment over and over again, switching doors every time, you will find that the more times you do the experiment, the closer your winning proportion will get to 2/3.

**ross** on October 16, 2019:

After you have made a choice, you will then be shown a wrong choice, after which you can choose again. So in every instance the probability of choosing a car is 1/3. on your first choice and 1/2 on your second choice.