The Monty Hall Problem is a big source of consternation. Many a math enthusiast has been confounded by the surprising, almost mystical doubling in probability from a seemingly innocuous change in choice.
But I’m getting ahead of myself. So let’s start from the beginning.
Imagine you’re on a gameshow. In front of you are three large doors, and the gleaming-toothed host, Monty Hall, tells you that behind one of those doors is a brand new car, and behind the other two doors are goats. You get to choose the door, and you win what is behind the door.
Assuming you want the car, the probability of winning is rather straightforward: 1 outcome in which you win, out of a total of 3 outcomes, makes a probability of 1/3. You can try out the game below as many times as you like.
[interactive — simple Monty Hall example]
You’ll notice that the more you play the game, the closer your win percentage gets to 1/3, or .33. This would be a pretty simple probability problem, but alas, this isn’t actually the Monty Hall problem of lore.
In the real Monty Hall problem, after you make your choice of door, the host does not immediately show you what is behind. Instead, he chooses one of the other doors, and reveals what is behind. He always chooses a door with a goat behind it.
[image — host revealing goat behind door]
The host then gives you a choice: you can open the door that you originally chose, or you can switch to the other remaining door and open it.
“Why would you want to switch doors?,” you might ask. Each door has the same probability of having the car, after all. Here’s the surprising thing: your chances of winning a car double to 2/3 when you choose to switch doors.
This result can throw your average math enthusiast for a loop when they hear it for the first time. Paul Erdős, one of the most prolific mathematicians of the 20th century, famously doubted the result (at least at first).
You can try out the way the Monty Hall game works below — once again, as many times as you like:
[interactive — the real Monty Hall game]
So, is there something magical about the host opening a door that moves the car? Well, no. And the reason for this has to do with what probability and chance actually are.
It’s true that, in each game of Monty Hall, there’s no winning 2/3 of a car or 1/3 of a goat — you either win or you lose. The probability in this context refers to how often the players win.
You can see this probability play out as you play the game many times. When you consistently choose to switch doors, you’ll notice that you win more often than you lose.
It might not be obvious why this is the case, so here’s the same game again from a different perspective: the host’s. Here, we can see where the car and goats are from the start. The computer-controlled player will pick a random door, and then you’ll have to open a door that contains a goat. Then, the player will always switch.
[interactive — Monty Hall game in which the car is revealed from the start]
Unfortunately, the computer-controlled player will always lose when they pick the right door from the start. But that only happens 1/3 of the time. The other 2/3 of the time, when they pick the wrong door from the start, there’s something special that happens — did you notice what that is?
[gif — Anchorman]
That’s right. 2/3 of the time, it works every time.
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