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The Monty Hall Problem

An American game show left an unexpected legacy: many arguments, and more than a few Web pages. Some people even learned some probability theory. We'll leave out the theory here to concentrate on different ways to understand the problem's solution.

The Problem

The game show Let's Make A Deal, hosted by Monty Hall, ended each show the same way. There were three closed doors. Behind one was a prize, while the other two concealed booby prizes.

Monty asked the contestant to choose a door.

Then Monty opened one of the remaining doors, revealing a booby prize.

Monty then offered the contestant the option to stay with the originally chosen door or switch to the other unopened door. The contestant received whatever was behind the chosen door.

Now for the big question: is it better to stay, better to switch, or does it make no difference?

The Importance of Empiricism

If you think it doesn't make a difference, stop. Right now, and I mean right now, you're going to play the game. Use this Java applet, read some game transcripts, or play it manually.

To play manually, you'll need:

• A friend
• A deck of cards
• Pencil and paper

Here's how to play:

1. Pick three cards out of the deck, two black and one red. Put the rest of the deck away. Make sure you see the cards at the beginning, so that you're sure you've got the right number of each color. (It's probably not a good idea to play with a magician, or someone skilled at card tricks or poker.)

2. On the paper, make four headings: Stay-Win, Stay-Lose, Switch-Win, and Switch-Lose.

3. The friend shuffles the cards and holds up all three, faces hidden from you.

4. You choose a card.

5. Your friend shows you one of the two cards you didn't choose. It will be a black (losing) card.

6. Announce whether you want to stay with your original choice, or switch to the other unrevealed card.

7. Your friend shows you the card you just chose. If it's red, you win; black, you lose.

8. Make a mark under the appropriate heading. For example, if you switched cards and lost, make a mark under Switch-Lose.

9. Repeat steps 3-8 at least several dozen times. The more repetitions, the better the statistics you'll get. Forty repetitions should be a minimum.

Once you're done, total the marks under each heading.

Okay, go play the game right now. I'll wait.

So... do you still think the odds are 50/50?

If you do, I bet you didn't play the game. "Why should I?" you think. "I know how it will turn out."

If that's your thinking, go play the game.

Still not convinced? Go play the game.

Why Does Switching Matter?

By now you've played the game and learned that it does indeed matter whether you stay or switch. However, why this is so might not be clear to you.

If you remain convinced that the odds are even whether you stay or switch, go play the game.

An Important Lesson

The more I consider and discuss this problem, the more obvious one fact becomes: most people will vociferously defend their beliefs despite having no evidence. Even though they have nothing to back up their answer besides their intuition, some people will argue at great length that it doesn't matter if you stay or switch. Indeed, some people will even refuse to play the game, claiming that their belief is so obviously correct that it could not be wrong!

That's an important lesson. Encountering this behavior made me raise my own standard of evidence. I've also learned that in situations like this, the first response should be go play the game. First get some evidence, then talk to me.

Addendum, 2005-06-02. The June 2005 issue of Wired offers another example of this problem in an article about a counterintuitive European method of firefighting. The technique has proven its effectiveness in fighting certain kinds of fires (e.g., a 50% drop in firefighter deaths over thirty years in Sweden, a 100% drop in Britain over twelve years), yet adoption of the technique in America has been very limited. The kicker was finding out that a journal of firefighting published a three-part article disparaging the technique—yet the article's authors admitted they had not been trained in the technique before they tested it. So how do they account for the drop in firefighter deaths in other countries?

A Modest Proposition

This page used to hold an offer to play 30 rounds of this game via email, for money. The offer was posted for more than three years before an astute visitor noted that the game's payoffs weren't in my favor. I'd written a set of Web pages explaining a widely misunderstood probability problem, yet miscalculated the probabilities for the payoffs of a very similar (but not identical) problem! Hoist with my own petard, as they say.

The effort wasn't a complete loss: I did get some practice creating a protocol that uses encryption.

No one ever took the offer. But then, how many people would actually go play the game?

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