I know, it sounds like the start of a lame joke. Actually, I’m talking about the classic Monty Hall Problem*, aka, Marilyn and the Goats. In my quest to catalog for you everything in the world that is interesting, I noticed I’ve not yet spoken about that car, let alone those goats!

I’ll set the stage for you. There’s this old game show called “Let’s Make a Deal”. On this show are three closed doors which face a contestant. Behind one of these doors is a car; behind the other two are plain ol’ goats. The contestant does not know which door leads to the shiny new car, but the game show host does.

And we begin. The contestant designates a door. But wait! Before that door is opened, the host surprises us by opening one of the other two doors to reveal… a goat. The contestant is now given a strange choice: he could stay with the original door, or switch to the other unopened door.

What should the contestant do?

If you’ve not heard of the Monty Hall Problem before, this might sound like a very silly question. As we all know, the contestant had a 1/3 chance of selecting the car. How does opening a door after the selection change the odds? Either by staying with the original choice or by switching to the new door, the odds should logically be the same. Right?

The self-proclaimed “genius”, Marilyn vos Savant, began one of her weekly columns back in September of 1990 describing the goat and car problem to readers all across America. She claimed in her response that the odds of choosing the car DOUBLE if the contestant switches to the other unopened door. She was inundated with letters from around the country declaring that, at last, the guru had messed up.

The thing is, she didn’t mess up. She is quite right.

If you’re a statistics student, you’ll likely see this as a classic case of Conditional Probability, where P[A|B] = P[A and B]/P[B]. In fact, this professor shows how conditional probability can be used to solve the Monty Hall problem.

But I’ll show you in a less mathy sort of way.

Let’s say you pick door one. This game has only three outcomes. Here they are:

door one door two door three
Outcome 1 CAR goat goat
Outcome 2 goat CAR goat
Outcome 3 goat goat CAR

Now let’s play out the contestant’s two options: to switch or not to switch. Clearly, if the contestant does not switch, he will win only once (in Outcome 1), which is 1/3 of the time. Now watch what happens if he switches. In outcome 1, the contestant had picked the door with the CAR. He switches to a door with a goat and LOSES! However, In Outcome 2, he’ll be shown door three, because it’s the only one of two and three that has a goat. The contestant switches to door two and HUZZAH! he wins a CAR. Likewise in Outcome 3, he’ll be shown door two, he’ll switch from door one to door three and WHAMMO! he wins a CAR. That means that with the strategy of switching, the contestant did in fact DOUBLE his odds of winning. He’ll win 2/3’s of the time. Interesting, eh?

Here’s another wacky problem for you. It’s called The Missing Dollar

Three men go to stay at a motel, and the man at the desk charges them $30.00 for a room. They split the cost ten dollars each. Later the manager tells the desk man that he overcharged the men, that the actual cost should have been $25.00. The manager gives the bellboy $5.00 and tells him to give it to the men.

The bellboy, however, decides to cheat the men and pockets $2.00, giving each of the men only one dollar.

Now each man has paid $9.00 to stay in the room and 3 x $9.00 = $27.00. The bellboy has pocketed $2.00. $27.00 + $2.00 = $29.00 - so where is the missing $1.00?

And, just for fun, here are three more classic puzzles:

Q. A prisoner is told “If you tell a lie we will hang you; if you tell the truth we will shoot you.” What can he say to save himself?
Q. Looking at a picture a man says, “Brothers and sisters I have none, but that man’s father is my father’s son.” Whose picture is it?
Q. What is the eleven letter word that all Yale graduates spell incorrectly?

Enjoy!

* Mark Haddon’s novel, the curious incident of the dog in the night-time, talks about this problem from the eyes of a young mathematical savant. It’s a good read.

Podcast