The Monty Hall Problem

Introduction

The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975. It became famous as a question from reader Craig F. Whitaker's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990.

Problem

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?

Solution

Since there are two doors left, with a car behind one and a goat behind the other, most people believe that the probability that the car is behind your selected door is $\frac{1}{2}=50\%$. But this would only be true if the probability that the car was behind each door was the same, and this is the incorrect assumption that most people make.

When you're told to make your initial choice, the probability that your chosen door hides the car is $\frac{1}{3}\approx 33\%$. Likewise, the probability that the car is behind one of the other two doors is $\frac{2}{3}\approx 67\%$. The key thing to notice here is that these probabilities don't change when Monty Hall opens one of the doors. You still have a $\frac{1}{3}\approx 33\%$ chance that your door hides the car, so there is still a $\frac{2}{3}\approx 67\%$ chance that it is elsewhere. Previously, elsewhere referred to the 2 doors that we didn't choose. But after Monty Hall opens one of them, that elsewhere now refers to the only remaining door we didn't choose.

The result is that there is a $\frac{1}{3}\approx 33\%$ chance that the car is behind our door, and a $\frac{2}{3}\approx 67\%$ that it is behind the other remaining closed one. Because of this, if we want to maximize our chances of winning the car, we have to switch!

If this still seems hard to believe, consider an extreme version of this problem. Let's say there are now 10 doors, with a car behind one of them and a goat behind each of the other 9. After you make a choice, Monty Hall opens 8 doors with goats behind them, and asks you if you want to switch. Even though there are once again only two remaining unopened doors, with a car behind one and a goat behind the other, there is only a $\frac{1}{10}=10\%$ chance that you made the right choice initially, it is much more likely that the car is elsewhere.

Conclusion

If you had (or still have) trouble understanding why switching is the better choice, don't worry, you are not alone. When first presented with this problem, an overwhelming majority of people believe switching does not matter, including people with PhDs and even Nobel physicists. Calculating probabilities is hard, but like with most things, the best way to get better is to practice, so don't give up!