The Monty Hall Problem

Updated on April 9, 2020
David3142 profile image

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

Monty Hall: The Host of 'Let's Make a Deal'

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?

The Three Doors. Here We Have Chosen Door 2 and Door 1 Has Then Been Opened to Reveal a Goat. Should We Switch to Door 3?

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, 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.

Monty Hall Problem Prizes


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!

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, so it is the probabilities of the original three doors that is important.

The Monty Hall Problem Explanation Video

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.

Three Options of Car Placement


In the picture above we are looking at what could happen if your original choice of door 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


    0 of 8192 characters used
    Post Comment
    • David3142 profile imageAUTHOR


      5 months ago from West Midlands, England

      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.

    • profile image


      9 months ago

      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.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
    ClickscoThis is a data management platform studying reader behavior (Privacy Policy)