Mathemagical
It can make Brad Pitt's buttocks wiggle and explain the death of fish. Meet mathematics, queen of the sciences.
A man, about a dog
Alvin Hall heads down the greyhounds to compare gambling with other investment opportunities. Is it possible he'll be beating the bookies?
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Parade, a popular American magazine, has a section called ‘Ask Marilyn’. Readers send in questions on all kinds of topics, and Marilyn vos Savant, occasionally billed as “The World’s Most Intelligent Person”, answers them.
In 1990, she received the following question from one Craig F. Whitaker:
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 number 1, and the host, who knows what’s behind the doors, opens another door, say number 3, which has a goat. He says to you, “Do you want to pick door number 2?” Is it to your advantage to switch your choice of doors?
You might prefer the following argument. Initially, there’s one chance in three (probability 1/3) that the contestant chose the right door, and two chances in three (probability 2/3) that they didn’t. Whatever the contestant chose, the host can open a door with a goat behind it, so the fact that the host did this does not affect those probabilities. So the contestant can stick with the original choice (door 1), and they’ll still have a probability of 1/3 of winning the car. Or they can change and say that the car is behind one of doors 2 and 3. What the host’s action has told them, that the contestant did not know before, is which of the remaining doors might have the car behind it. It can’t be behind door 3 now, because that has a goat. So the 2/3 probability of getting the car, that originally applied to doors 2 and 3 taken together, now applies just to door 2, and the contestant should switch.
Still not convinced? Some people find it useful to think of a rather different version of the game, as follows. Suppose that instead of three doors there are 100 doors, with 99 goats but still only one car. You pick a door. Your chances of winning the car are very small, just 1 in 100. The host generously opens 98 of the doors you didn’t pick, every one with a goat behind it. This leaves two doors still closed, your original choice and one other. Now should you switch?
Still unconvinced? The very informative article on the Monty Hall problem in the Wikipedia online encyclopaedia has several other arguments that might persuade you, some of them involving more formal uses of probability. Some people are convinced by computer simulations of the problem — here's one of several that you can try online.
Thinking about the problem in terms of probability does help to show how important it is to be clear about the assumptions behind a question like this. We assumed, for instance, that the host would always open a door with a goat, regardless of your original choice, and would always offer you the chance to switch. But suppose you’ve watched this game show for years, and you know that actually the host opens the goat door and offers the switch only to contestants whose original choice of door had the car behind it. Well, if that’s really the case, you’d be crazy to switch.
Finally, a bit more American media history. You might be wondering why this is called the Monty Hall problem. Monty Hall was the real host of a popular, long-running real American game show called “Let’s Make a Deal”, that was broadcast between 1963 and 1991, and has also reappeared in various forms since then. The show even has a website at www.letsmakeadeal.com .Let’s Make a Deal did involve ‘prizes’ that you wouldn’t much want to win, as well as valuable ones, and Monty Hall did offer the contestants opportunities to change their choices in various ways, though apparently it never ran a contest that was exactly like the one we’ve been discussing. Versions of the show have been broadcast in many countries other than the USA, though apparently not in the UK.
And the Monty Hall Problem itself dates back to a lot earlier than 1990. It seems to be generally accepted that its first appearance was in a letter by the statistician Steve Selvin, published in the journal The American Statistician in 1975, though in that version there were boxes instead of doors, and the ‘bad’ boxes were simply empty rather than containing goats. In a follow-up letter, Selvin reported that “several correspondents claim my answer is incorrect.” Since The American Statistician is published by the American Statistical Association, most of its readers are professional statisticians, so this shows that the history of learned people getting confused about game shows is also a long one.
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Content last updated: 12/12/2005
About our expert
Kevin McConway is a Senior Lecturer in the Department of Mathematics and Statistics at the Open University, where he teaches statistics and health studies, and researches in several areas including statistical theory, health service organization, ecology and evolution.
He has degrees in mathematics, statistics, psychology and business from the Universities of Cambridge and London and the Open University. Kevin originally comes from rural Northumberland but is now a long-term Milton Keynes resident.
