Saturday, December 3, 2016

The Monty Hall problem: Amy Santiago is right. You should switch doors.

Oh Amy, you're such a funky cat with your feisty stats, and I love you for that.

Let’s talk about the Monty Hall problem, as featured in the most recent episode of my favorite TV show of the moment, Brooklyn Nine-Nine.

In the episode, the precinct’s captain, Captain Holt, is in an argument with his husband over the Monty Hall problem, and one of the detectives, Amy Santiago (who, incidentally, is my fictitious dream woman, because she is wonderful and adorable and I am apparently a dude who crushes on TV characters), is trying to figure out a way to explain the correct answer.

The Monty Hall problem, named for the original Let’s Make a Deal host Monty Hall*, is as follows: You’re on a game show trying to win a car, and the host asks you to pick one of three doors. Behind one of the doors is the car; the other two contain a goat. After you pick a door, the host—who knows which is the car door and which are the goat doors—will open an unpicked door revealing a goat and ask if you want to stay with the door you picked or switch to the unopened door you didn’t pick. Should you switch doors?

The answer is indisputably yes, you should switch. By switching, you double your chances of winning the car—staying gives you a 1 in 3 chance of winning the car; switching improves your chances to 2 in 3. It may be counterintuitive, but you can test this out by playing repeated simulations of the problem, either in real life or online. The result, with enough simulations, will be the same: if you stay, you’ll win the car about 33 percent of the time; if you switch, you’ll win about 67 percent of the time.

Holt is unconvinced, thinking that, once it’s down to two doors, it’s simply a 50/50 chance and switching doesn’t affect anything. Amy, who’s the sort of person who gets excited over weekend math conferences called Funky Cats and Their Feisty Stats (again: fictitious dream woman like whoa), is determined to explain why switching is the correct strategy.

Unfortunately, we never get to hear Amy’s explanation because it turns out the spat wasn’t really over the math problem but rather a dearth of boning. So let me take a stab at it.

In the Monty Hall problem, there are exactly nine scenarios that can play out: you picking Door 1 and the car being behind Door 1, 2, or 3; you picking Door 2 and the car being behind Door 1, 2, or 3; and you picking Door 3, and the car being behind Door 1, 2, or 3. They're organized in this chart:

As the chart indicates, you win the car by staying in only three of the nine scenarios. You win the car by switching in six of the nine scenarios. Thus, staying yields only a 3 in 9 (i.e., 1 in 3) chance of winning; switching yields 6 in 9 (i.e., 2 in 3) chance of winning. Thus, you should switch—it doesn’t guarantee the car, but it significantly improves your chances.

Another, perhaps less intuitive, way of thinking about it: let’s say you pick Door 1. When the host asks you to switch, he’s asking if you’d like to change your pick from "Only Door 1" to "Either Door 2 or Door 3"—in other words, asking if you’d like to change your chances from "1 in 3" to "2 in 3." The chance of your door being a car doesn’t magically increase once the goat door is opened; it remains 1 in 3, so the alternative choice is 2 in 3.

It becomes a little bit clearer if, instead of three doors, we’re playing with a million doors. When you first pick a door, you have a 0.0001 percent chance of winning a car. When the host opens 999,998 doors revealing 999,998 goats, you’re left with your door and, say, Door 784,912. You intuitively know to switch—it’s highly unlikely that you picked the right door on your first try, and Door 784,912 just looks so appealing sitting there all by itself. In essence, the offer to switch is asking if you’d like to stick with your 0.0001 percent chance of winning, or if you’d like to take the 99.9999 chance that your initial pick was wrong. Even more simply, the offer to switch is asking, "Do you think the car is behind the one door you picked, or any of the other 999,999 doors?"

With a million doors, switching makes sense. And the math stays true with 500,000 doors. And 10,000 doors. And 50 doors. And indeed, all the way down to three doors.

If the whole doors thing makes this a bit opaque, consider this mathematically identical problem: let's say I'm thinking of a (whole) number between 1 and 1,000,000. Once you pick a number, I tell you that the number I'm thinking of is either the number you picked or 784,912, and I ask you if you want to switch to 784,912. Of course you'd switch—intuition dictates that it's highly unlikely you just happened to pick the correct number out of a million, and switching just makes sense. Clearly, once you're down to two numbers, switching does affect your chances significantly; you're going from having a 1 in 1,000,000 chance to a 999,999 in 1,000,000 chance. The concept remains the same if I'm thinking of a number between 1 and 1,000,000 (99.9999 percent chance of winning if you switch), 1 and 10 (90 percent chance), 1 and 5 (80 percent chance), and, yes, 1 and 3 (about 67 percent chance).

Finally, if all of that is still unconvincing, here's maybe the simplest explanation. If you stay with your door, the only way you can win the car is if you initially picked the correct door, which has a 1 in 3 chance of happening since there's only one correct door. If you switch, the only way to win is if you initially picked a wrong door, which has a 2 in 3 chance of happening since there are two wrong doors. Basically, you probably picked the wrong door initially, so staying means you're probably going to stay with the wrong door, while switching means you're probably switching to the correct door.

In conclusion, you should definitely switch doors because Amy is right. Also, Amy is totally dreamy, even when she's shame-eating hamburgers.

God, I need a girlfriend. Or Season 3 on DVD, either one's fine.

*It’s worth noting that, on Monty Hall’s Let’s Make a Deal, this situation as described never came up on the show, and thus, the name of the problem is kind of a misnomer. It's like having a problem about funny jokes, and calling it the Big Bang Theory problem. Bazinga! (For real, that show sucks.)

On the other hand, Monty Hall himself was able to come up with the right answer to the problem pretty quickly—a lot faster than many PhDs and other math experts, in fact—so maybe he deserves to have the problem named after him after all.

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