The Monty Hall Problem

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?

The official answer is…switch. You double your odds of being right by switching.

This has led to some vigorous debate.

I read it in Jo Craven McGinty’s final column, instantly decided all of these idiots were wrong, and did some Matlabbery.

The idiots were right. You double your chances of being right by switching.

BUT WHY?

Let’s go chronologically.

Step 1:

The car is somewhere, the goats are somewhere. These positions are fixed. No car-goat switching shenanigans take place.

Step 2:

You pick a door.

You have 1/3 chance, 33.33….% chance of getting it right.

THAT IS IT. THIS GUESS IS DONE. THE CHANCE OF YOUR INITIAL GUESS BEING CORRECT IS FINISHED, COMPLETED, SET, and NOT CHANGING.

Step 3:

The host looks behind the doors and picks one with a goat. This door is opened. You, the contestant, can see the hosts door opened to reveal a goat.

Step 4:

You may change your guess to the other door.

This is where the magic happens.

That initial guess is locked at 1/3 chance, 33% (and hereafter I’m neglecting the .333… repeating). It cannot change. But there’s still an outstanding 2/3 chance of finding the car. The car has to be somewhere. Since you already know, for a fact, that there’s only 1/3 chance of your first guess being correct, and you know there’s 0% chance of it being behind the host’s door (remember, the host opened that door. You can see the goat), the remaining chance has to be behind the other door.

So switching has to improve your odds.

But it just doesn’t feel right, does it? Why not?

Because you think about things after they’re done.

If you picked your first guess after the host opened a door, then the two options would be 50/50, what feels right. But that’s not what happens. The first guess is made and locked BEFORE the host opens the door, so it has to be fixed at 1/3. The host doesn’t move the cars and goats around after opening a door, so the odds don’t reset.

Matlab: MontyHall
(Change ending from .txt to .m or copy and paste)

I suppose you also might want a goat. I’m neglecting that.