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Not Monty Python... the other one

Monday, March 10, 2025 8:01

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By Brian Clegg

A book I’m currently reading (Proof by Adam Kucharski) starts by introducing the Monty Hall problem and adds a piece of info I wasn’t aware of. It’s my favourite example of the counter-intuitive nature of probability, featuring at some length in my book Dice World, so I thought it worth revisiting here.

The problem (of which more in a moment) gained worldwide fame when it featured in the ‘Ask Marilyn’ column in Parade Magazine in 1990. The column was written by Marilyn vos Savant, whose claim to fame was appearing in the Guinness Book of Records as the person with the highest IQ in the world. (She was born Marilyn Mach, but despite appearing to have a phoney attempt to get the word ‘savant’ into her name, vos Savant was her mother’s surname.) What was remarkable about the problem was that so many people – some of them mathematicians and professors – got it wrong. It is fascinating now to look back and see some of the letters published in Parade saying what a terrible mistake vos Savant made.

The thing I discovered in Proof is that that the great mathematician Paul Erdós would not initially accept the correct answer because the person describing the problem to him could not adequately explain why it is the correct answer. The mathematician believed that for something to be proved it had to be explained too. I’ve tried various explanations over the years, but I added one in my book Game Theory, so I wanted to bring that into the mix – and also to make use of a live version of the game for your entertainment.

If you already know the problem, you might like to skip the next bit. It is based on the game that ended the quiz show Let’s Make a Deal, hosted by Monty Hall. In the version described by vos Savant, the winning contestant is given a choice of three doors. Two have goats behind them, one has a car. It’s a purely random choice, so when the contestant picks a door – say door 2 – there is a 1 in 3 chance they are right and a 2 in 3 chance they are wrong. The game show host now opens one of the other doors and shows a goat. Finally the contestant is given a choice. Would they like to stick with the door they have, or switch to the other unopened door? The question is, should they stick, should they switch, or does it not matter (probability wise) which they do?

Here’s the game in action:

My audience was unusual, but the vast majority of people can see this is very simple. There are two doors available (because we can discount the one the host opened with a goat behind it). One has a goat, one has a car. So it’s 50:50 which will be right. This means it doesn’t matter if you stick or switch.

The vast majority of people are wrong. You will double your chances of winning if you switch.

Here’s one explanation of why. Remember at the start, there was a 2 in 3 chance the car was behind one of the other two doors. All the game show host does is show you which of those two not to choose – but there is still a 2 in 3 chance the car is there. So you ought to switch. This only works because the game show host has information you don’t. He selected a door he knew to have a goat behind it.

Here’s the game theory take on the Monty Hall problem, assuming that the player initially selects door 1, which is how the problem was first posed. If the player switches, they will always switch to the door the host didn’t open, as otherwise they are definitely switching to a goat:

If, despite these arguments, you don’t find it convincing you are not alone. I remember when the problem was first publicized many of us wrote little computer simulations to prove the outcome. And it’s right. Switch and you have a 2 in 3 chance of winning. But, as I mentioned, the most fascinating thing were the irate letters vos Savant received and reproduced. Here are some of my favourites (I have replaced names with initials to avoid any blushes). I particularly love the last one:

I’ll come straight to the point… you blew it! [repeats the problem] Let me explain: if one door is shown to be a loser that information changes the probability to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and, in the future, being more careful. – R. S. PhD, George Mason University, Fairfax, Va.
You blew it, and you blew it big! I’ll explain: After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your answer or not the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame! – S. S. PhD, University of Florida
I have been a faithful reader of your column and have not, until now, had any reason to doubt you. However, in this matter, in which I do have expertise, your answer is clearly at odds with the truth. – J. R. PhD, Millikin University
May I suggest you obtain and refer to a standard textbook on probability before you try to answer a question of this type again? – C. R. PhD, University of Florida
Your logic is in error, and I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns. – W. R. S. PhD, Georgia State University
You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed toward the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind? – E. R. B. PhD, Georgetown University
You’re wrong, but look at the positive side. If all those Ph.D.s were wrong, the country would be in very serious trouble. – E. H. PhD, US Army Research Institute

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Now Appearing is the blog of science writer Brian Clegg (www.brianclegg.net), author of Inflight Science, Before the Big Bang and The God Effect.

Source: http://brianclegg.blogspot.com/2025/03/not-monty-python-other-one.html

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