The Monty Hall Problem

Sometimes I think so much that I run out of my own thoughts. It’s rare, but it happens.

My brain doesn’t idle well, so I fill it with books and articles. Monty Hall and probability reviews have provided my brain with enough thought material for days.

The Monty Hall Problem

The Monty Hall Problem is the most famous problem in probability. That makes it about as well-known as the capital of Nebraska (it’s not Nebraska City…I know, I was surprised, too).

The problem gets its name from Monty Hall, host of the classic game show Let’s Make a Deal. You’re faced with 3 doors. Behind one door is a new car. A goat is behind the other two doors. You pick one of the 3 doors. (For purposes of this problem, assume you win if you pick the car and not a goat.)

After you pick a door, Monty Hall opens one of the other two doors to reveal a goat. Two doors remain. He asks you if you want to switch your choice from your original pick to the remaining door.

The question is, should you switch?

Double Your Fun

I first encountered this problem and thought it didn’t matter if you switched. The probability was the same at 50%, as two doors remained. Most people think similarly.

Most people are wrong. Switching increases your odds of success (again, assuming you are not a goat enthusiast) from 1/3 to 2/3. What??

Let’s take it from the top. When we start, there are 3 unopened doors, one prize, and two decoys. We make our choice, and our odds of winning are 1/3. So far, so good.

It’s when a decoy door is opened and we’re offered the chance to switch that probability cuts across our natural intuition. It’s easy to be fooled into thinking because two doors remain, probability is 1/2.

Understanding why switching is better is a bit clearer if we change the order order of things slightly. You make your pick. Now suppose that instead of opening a door, the host offers you a chance to switch and take the other two doors. Obviously that’s a good deal, two doors are better than one.

Well, that’s basically what the host is offering you: a chance to take two doors. It’s just that one of them happens to be opened already, and there was a goat behind it. But even if the door wasn’t opened, and you could switch to two unopened doors instead of switching to one unopened and one opened…there was always a goat behind one of those doors anyway.

The host always opens the door with the goat. So in effect, there is no difference between the scenarios. And switching doubles your odds.

Brain Drain

If you’re still not seeing the logic, I don’t blame you. This video does a better job of explaining it than I do.

I’m going to sign out of this article before I overthink it and get confused again. But I have one more thought before I go: the brain fascinates me. Especially because there are so many blind spots in its mechanics. As a monkey mind zookeeper, knowledge of these blind spots is useful. It helps me monitor the monkey’s tricks so he doesn’t start breaking things again.

And it reminds me that I’m not alone. Anyone with a brain can be fooled. It’s a hardware issue!