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?
Well, what do you think?
Before revealing Marilyn’s answer and discussing the problem a bit more, let’s clarify one aspect of the question. The answer depends on how you think the game show host behaves. Does he always show contestants what’s behind another door? Perhaps he does this only if they originally chose the door with the car. But let’s suppose not. Let’s suppose that he does always open another door and offer the contestant the chance to switch their choice. Also, since the host knows what’s behind the doors, let’s suppose he never opens the door with the car behind it (because that would spoil the fun rather).
OK, with these extra suppositions, would you switch?
Marilyn’s answer was that the contestant should switch. To be more precise, she argued that, if you switch, you will have two chances in three of winning the car (that is, a probability of 2/3 of winning), but if you stick with your original choice, you’ll only have one chance in three of winning (that is, a probability of 1/3). In other words, if you switch, you still might not win, but you’ve improved your chances.
Maybe that’s what you thought. But maybe not. Many people think along the following lines. After the host has shown the goat behind door number 3, there are two doors left that the car might be behind. You still don’t know which, so there’s one chance in two that the car is behind door 1, your original choice, and one chance in two that it’s behind door 2, the choice you’ll make if you switch. So it doesn’t make any difference if you switch. You can if you like, but there’s no disadvantage in sticking with your original choice.
If that’s what you thought, you’d be in good company. The original Parade article generated a huge correspondence, including letters from professors who advised Marilyn vos Savant to stick to what she knew and not mess about with questions of probability. Since then, there have been articles and correspondence in the press in many different countries, heated discussions on Internet mailing lists for statistical experts, several articles in learned journals, and even a couple of books that address the problem. The car and the goats continue to crop up again and again in the media. For instance, after the economist John Kay described the problem in his column in the Financial Times in August 2005, a correspondence about it ran for a fortnight.
OK, why do your chances of winning increase to two out of three if you switch? There are several ways to explain this. I’ll go through a couple of them. However, I’m aware from all the previous discussions that none of these ways might convince you! Different people are convinced by different arguments, and some never are.
One way to think of the problem is in terms of three different scenarios. There are three ‘prizes’: the car, goat 1 and goat 2. The possible scenarios are as follows.
- The contestant originally picks goat 1. The host shows goat 2. The contestant will win the car by switching.
- The contestant originally picks goat 2. The host shows goat 1. The contestant will win the car by switching.
- The contestant originally picks the car. The host shows either one of the goats. The contestant will lose by switching.
These three scenarios are all equally likely (because the contestant has no idea where the car is, originally, and is making the original choice at random). In two of the scenarios, the contestant wins by switching. In the third, the contestant loses by switching. So the contestant has two chances in three of winning if they switch, and only one chance in three if they don’t switch.
Looking at it this way also shows what’s wrong with the argument that there are two possibilities after the host has opened the door — car behind door 1 and car behind door 2 — and that these are equally likely, so that it doesn’t matter whether you switch or not. There are indeed two possibilities, but they aren’t equally likely.
Convinced? Some people are persuaded by this; others can’t see why it makes sense to treat these three scenarios as equally likely.
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.