Puzzles

# The Monty Hall Problem Is the Probability Puzzle That Enraged 10,000 Readers

When you first hear it, the Monty Hall Problem sounds simple enough. You're on the game show "Let's Make a Deal!" and host Monty Hall presents you with three doors. One, he says, has a car behind it. The other two are hiding live goats. He asks you to choose a door, and you go with Door #1. He then opens Door #3 to reveal ... a goat. He now gives you the chance to either stick with your choice of door, or switch to Door #2. Should you switch?

The seemingly obvious answer is that it doesn't matter — you've got a 50-50 shot at winning the car either way. The correct answer, however, is that you should absolutely switch.

## "You're the Goat!"

This puzzle was first published in a letter to the editor of The American Statistician in 1975 by Berkeley professor Steve Selvin. But it didn't reach the height of fame until 1990, when the "world's smartest woman" Marilyn vos Savant answered a reader's question about the problem in her weekly column in Parade Magazine. The reader posed the problem the same way we did just now, and vos Savant answered simply: "Yes; you should switch," she replied. "The first door has a 1/3 chance of winning, but the second door has a 2/3 chance."

If this baffles you, you're certainly not the only one. In response to this simple column, she received more than 10,000 letters, many from leading academics, telling her she was wrong. They weren't exactly polite, either. "Since you seem to have difficulty grasping the basic principle at work here, I'll explain," wrote one Ph.D. "Maybe women look at math problems differently than men," wrote another reader. Yet another reader went straight for the grade-school approach: "You are the goat!"

But vos Savant was right, and here's why.

## Pick a Door, Any Door

The key to this puzzle is to realize that it isn't truly random: Monty Hall knows which door contains the car, and he's not going to open it until the end of the game. Imagine that instead of three doors, you're faced with 100. You choose Door #1, and Monty Hall opens 98 other doors to reveal a goat in every single one of them. Doesn't the one other door left closed feel kind of ... suspicious? Do you think the door you chose at the beginning is just as likely to contain the car as the only one left closed after opening 98 others?

Three doors feels less intuitive, but the odds work the same way. Before you choose a door, there's a 1/3 chance that any one of them contains the car. But once you choose Door #1 and Monty Hall opens Door #3 to reveal that it doesn't contain the car, Door #2's chances shoot up to 2/3. Remember, just like in the 100-door problem, there's likely something special about Door #2 that led him not to open it.

You can really understand it by breaking down the possibilities like Dr. James Grime does in the video below. We'll break them down here for you, too.

• Say you choose Door #1, and the car is in Door #1. Monty Hall opens Door #3 to reveal a goat. In this case, if you switch, you lose.
• Say you choose Door #1 and the car is in Door #2. Monty Hall opens Door #3 to reveal a goat. In this case, if you switch, you win.
• Say you choose Door #1 and the car is in Door #3. But Monty Hall wouldn't open Door #3 to reveal a car, because that ends the game prematurely. Instead, he'd open Door #2 to reveal a goat. Again, in this case, if you switch, you win.

As you can see, switching wins you the car two out of three times, because Monty is essentially giving you a choice to either open Door #1 only, or to open both Door #2 and Door #3. It seems incredibly counterintuitive, but that's probability for you.

For mathematical puzzles from the reigning king of mathematical puzzles, check out "The Colossal Book of Short Puzzles and Problems" by Martin Gardner. We handpick reading recommendations we think you may like. If you choose to make a purchase, Curiosity will get a share of the sale.

– singingbanana

### Key Facts In This Video

1. The traditional setup of the Monty Hall Problem places a goat behind two doors and a sports car behind one. 00:26

2. It's easier to understand the solution to the Monty Hall Problem if you envision 100 doors instead of 3. 01:53

3. The Monty Hall Problem is not genuinely random, which is why its solution is counterintuitive. 03:53

Ashley Hamer