The Monty Hall Problem
The Monty Hall Problem, as originally written, goes like this:
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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
It seems odd - surely the odds don't change? Yet, if you stick you'll only win a third of the games you play on average. Change your mind, and you'll win twice as many games.
It's important to realize that at the start, you've a 1-in-3 chance of picking the car. Or to put it another way, you've a 2-in-3 chance of picking a goat.
Now if you do start out picking a goat, then there's only one remaining goat I can reveal to you - this means that 2-out-3 times, I've got no choice in which box to reveal to you. This means, 2-out-of-3 times, you are better off switching because the only box I can leave unopened for you contains the car.
This site records all the games played. So far, we've logged 1063 games:
- Out of 334 games where the player didn't switch, they won 106 games (32%)
- Out of 729 games where the player switched, they won 474 games (65%)
- Keith Ellis has written a thorough article and includes some scholarly citations.
- Monty Hall problem: The probability puzzle that makes your head melt - BBC News article from Sep 2013.
- Monty Hall Problem on Wikipedia.
If want to comment on this, see my related blog post