The 'Monty Hall' Problem: Everybody Is Wrong

The 'Monty Hall' Problem: Everybody Is Wrong
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Every few years or so, the Monty Hall Problem has another moment in the sun. Just last week, Priceonomics brought it back again, in a post titled "The Time Everyone 'Corrected' the World’s Smartest Woman."

Here's the problem in its most famous formulation (most others are similar):

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 #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

Faced with this conundrum in 1990, Parade magazine columnist Marilyn vos Savant (the aforementioned World's Smartest Woman) responded that yes, you should switch, for doing so will boost your chances from 1 in 3 to 2 in 3. She was deluged with letters, some from very statistically informed people, informing her she was wrong.

She was wrong. So were her critics. The correct answer is, "What a terribly worded problem. Why don't you rework it and ask me again when it contains the information I need?"

The issue -- and I'm not the first person to notice this -- is that the problem doesn't give us enough details about the behavior of the host. The algorithm by which the host makes decisions -- whether to offer a switch, which door he opens -- needs to be filled in with the reader's assumptions, because numerous algorithms could lead to him acting the way that's described. Much of the debate here amounts to people who made different assumptions calling each other morons.

One possibility is that the host is trying to trick you to save the show money: He's offering you a chance to switch because you picked the car, and he wouldn't have offered it otherwise. In this case, switching is obviously a bad idea. Hilariously, this possibility is brought up and rapidly dismissed when the problem appears in the movie 21. ("My answer is based on statistics," the super-smart student explains. That settles it then!) Similarly, Priceonomics notes that, in the real-life show the problem is loosely based on, the host actually did reserve the right not to offer a switch -- completely missing that the problem doesn't rule this out either.

Another possibility is that the host is making his decision based on other considerations. Maybe he's acting more or less randomly to keep things unpredictable, or maybe he tries to steer the show in the direction of good TV. If this is the case, the best you can do is just try to read the situation.

Still another way to interpret the problem -- and the way vos Savant interpreted it -- is that the scenario isn't merely what happens the time you're on the game show; it's what happens every time: The host always opens a door, and that door always reveals a goat. Under this assumption, the odds that your original guess was right are 1 in 3, while you have a 2 in 3 shot at the car if you switch. This is because, in the 2 in 3 times you pick a goat to begin with, the host will helpfully reveal the other goat so you can pick the car.

The "correct" answer, this last one, is tricky enough to get when you start with the required assumption. Plenty of people have missed it even then. Those summarizing the problem in the future should make the assumption explicit.

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