Understanding the Monty Hall Problem

The Monty Hall problem is a logic puzzle named for the host of the gameshow Let's Make a Deal. It's one of my favorite such problems, because it's an example of math completely contradicting my gut instinct. Even though I know the math says to do one thing, my gut consistently says the other -- and thus every time the problem is explained, I get a little window into my fallible human brain. In fact, I used to be so tied to my gut on this one that I would fail to explain the Monty Hall problem correctly when describing it to others, because I always assumed my gut instinct was actually correct. (Yes, I took Math for Liberal Arts Majors.)

Marilyn vos Savant posed the problem in one of her columns like so:

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?

I won't spoil the answer. Think on it for a moment. Do you switch or not? Now watch this AsapSCIENCE video for a brief explanation of what math says is the right thing to do...and what my gut says is crazy:

Having said all this, it's important to note that we have a shared assumption that a car is better than a goat. But if we're talking about one of those awesome unicorn goats, maybe you want that instead.

Also relevant: previous coverage of Monty Hall, plus an interview with the man. And although they're not strictly related, the Monty Hall Problem occupies the same space in my brain as The Abilene Paradox.