You are on a game show, and get to choose between three doors. Behind one is a grand prize. Behind the others there's nothing. But there's a twist — and it can double your chances of winning. Welcome to the Monty Hall Problem.
Over the years the Monty Hall problem has become infamous. In a way, that has undercut its shock value. Most people know the answer before they get a chance to think about the question. To be fair, often knowing the answer is the only way to find the right way through the question, but it does take away some of the urgency.
The problem stems from a dilemma that quite a few people faced on the game show Let's Make a Deal. Monty Hall, the host of the show, gave people their choice of three doors. Behind one door was a valuable prize. Behind the other two doors were joke prizes - generally something like a goat. The person made their choice. Monty then opened one of the remaining two doors. The person then was given a choice. They could stay with the door they had chosen, or they could switch to the remaining closed door. (Or, I suppose, if they loved the goat, they could just choose the open door.)
Which move should a person make?
Contestants are twice as likely to win the grand prize if they switch their choice as they are if they stay on their original choice.
The Monty Hall solution gained fame when Marilyn vos Savant published it in response to a reader question in Parade magazine. It seemed incredible. Out of two doors, neither of which contestants had any definite information about, how could one door offer twice the chance of winning as the other door? The odds had simply gone from one out of three for all three closed doors to one out of two for both doors.
The most famous disbeliever was famous mathematician Paul Erdős. A Hungarian who had earned his doctorate at 21 and just kept working, he was a famous problem-solver. He came up with nicer, more elegant proofs of solved problems, he tirelessly sought out new people to collaborate with, he tried to get young people interested in math, and he offered cash prizes for newcomers who could come up with new proofs for unsolved problems. He was hardly a stick-in-the-mud, but it took him years to finally accept the by-then well-established solution to the Monty Hall problem. It was a computer simulation, which allowed him to play over and over, that convinced him. The problem can work against even the finest mathematical intuition. (Incidentally, Erdős gave himself the world's best epitaph. When asked what he wanted printed on his tombstone, he said, "I've finally stopped getting dumber.")
There are a lot of different ways to explain why switching is the better option. One way might just be visual, doing a quick sketch of three doors on a piece of paper, "hiding" the car between one of them, and then running through the results of guessing and switching.
Another involves slightly altering the problem. Instead of three doors, there are a hundred. You pick one. Monty opens up 98 doors, and you can either stay with your first guess or switch to the one other door left closed when the rest of the doors open. Would you stay with your original door in that scenario? This is a good way of thinking about it, because it shows you how much information Monty is giving you when he opens up those other doors. He can't arbitrarily open them. He has to make sure he doesn't reveal the prize. Unless you happened to pick that one door in a hundred that opens on to the prize, Monty has basically shown you where the grand prize is.
I think a good way of thinking about the Monty Hall problem is to run the repetitive simulations in your head, step by step. First, forget about the second step of the game. Forget that Monty is opening any doors. Just think of the game as you choosing one out of three doors, and then finding out if you won or not. What are the odds that you'll win, and what are the odds that you'll lose? You'll win one out of three times, and lose two out of three times.
Let's look at the two out of three times that you've picked the wrong door, and reintroduce the second part of the game. In these two cases, Monty will open the other losing door. He has no choice. He can't show you the prize directly. That means that, in two out of three cases, the remaining door is the winner. In two out of three cases - if you switch, you win.
Now let's look at the remaining case, that one time out of three when you guessed right initially. Monty can open any door, because they're both losers. In this case, yes, if you switch you will lose. But it only happens one third of the time. In one out of three cases - if you stay, you win.
So, in two out of three cases, switching your guess will win you the prize. In only one out of three cases, not switching your guess will win you the prize. If you switch, you double your odds of winning.
If you're still in doubt, you can play both strategies here, at Stay or Switch. It tallies up your win rate, and shows you the win rate of everyone, depending on their strategy.