Monty Hall Solution

By Thomas Simpson

If you haven’t seen the original problem from the start of Easter term, take a look at it here.

The Monty Hall problem was originally posed in the 1970s in the American Statistician and has become a popular maths puzzle in the years since.

There has been much debate over the correct answer but it has been shown both mathematically and via computer simulation the best option is to:

Click to show solution
Switch

If you don’t agree then you’re in good company – many mathematicians have found issue with the answer in the past and the exact mechanics of the game may differ from our simplified set-up.

A simple way of thinking of the solution is all the possible arrangements of the prizes. Let us assume you picked door 1 (with no loss of generality). Then we can look at all the possible set-ups of the problem:

Door 1Door 2Door 3Result without SwitchResult with Switch
GoatGoatCarGoatCar
GoatCarGoatGoatCar
CarGoatGoatCarGoat

The important realisation is that Monty will only ever open a door that doesn’t have the car behind it. From this, we can see you have a 2/3 chance of winning the car if you switch – but only a 1/3 chance if you stick.

This is quite a simplified way of thinking about the problem – if you’re interested in finding out more you can take a look at the Wikipedia article here which goes into large amounts of detail. In particular, take a look at the sections on Bayes Theorem where you can see a deep dive into the true probabilities behind the problem.

Bayes Theorem is of particular relevance now – it deals with conditional probabilities (knowing one thing, what are the odds of something else). With the amount of tests being conducted daily at the moment, Bayes Theorem is used to understand how many false positives may occur. Again, you can look at this Guardian article for a deep dive into this.

If you’ve got any questions about the problem or solution, don’t hesitate to comment or reach out to us on social media – you can find us on FacebookInstagram and Twitter. If you’ve got any suggestions for problems for the future, please do let us know.

And as a final thought – if you stuck with your first choice of door, you can always say you wanted to win a goat!

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