The Monty Hall Problem

If you're not familiar with the Monty Hall Problem, go read about it, say, here, because if this post is successful it will spoil the problem without instilling any of the wonder, perplexity, confusion, and amazement that should result from your first exposure to the problem.

Okay? Okay.

This blog post describes a fictional scene that takes place in one of those TV producers' rooms, with lots of televisions and cool electronic equipment and comfortable chairs.

Monty Hall sits in the room with a producer of Let's Make a Deal in front of a television screen.

Producer: Look, Monty, you're having this same long exchange with the contestant on every single show, and I have some ideas about how to change it to make it better television. Take a look.

The producer starts a clip on the television screen in front of them of a previous episode of Let's Make a Deal. In the clip, Monty Hall is standing, microphone in hand, with a Contestant in front of 3 doors. The clip plays:

Monty (on the television): There are three doors here, door number 1, door number 2, and door number 3. Behind one of these doors is a prize, behind the others nothing at all. You can pick one of these doors, and if the prize is behind that door then you have won it. Otherwise, you win nothing at all. So, Contestant, which door will it be?

Contestant: I'll pick door number 1.

Monty: Door number 1? Now, before I open it, I'm going to offer you a deal. You could get what's behind the door that you've chosen, door number 1, or you could get what's behind both of the other doors. Would you rather stick with door number 1, or would you rather get both door number 2 and door number 3?

Contestant: I believe I'll switch to the two doors.

Monty: You're going with doors 2 and 3? Now, you know that behind at least one of those two doors is nothing at all. Do you still want to switch?

Contestant: Yes, I'll stick with the two doors.

Monty: In fact...

Monty's lips can be seen to mouth "tell me an empty one" as he whispers into a hidden microphone. After pausing to listen to a hidden earpiece, he continues:

Monty: ... there is nothing at all behind door number 2. Do you still want to switch?

Contestant: Yes, I'll stick with the two doors.

Monty: Look, I'll show you. Open door number 2.

Door number 2 opens, revealing that there is nothing behind it.

Monty: Do you still want to stick with doors 2 and 3?

Contestant: Yes, I'll stick with the two doors.

Monty: Well, there is nothing behind door number 2, so I'm really only giving you what's behind door number 3. Do you still want to stick with door number 3 instead of going with your original choice, door number 1?

Contestant: Yes, I'll stick with door number 3.

Monty: So, even after seeing that there's nothing behind door number 2, you want to abandon your original choice, door number 1, and take what's behind door number 3?

Contestant: Yes, I'm going with door number 3.

The producer stops the tape.

Producer: Look, I think that it's interesting how you get the contestant to change his mind, but I think that we can tighten up this exchange and make it more interesting, to both the casual viewer and the educated viewer who might comment on our show. Why don't you act out with me this little script I wrote up. You be you, I'll be a contestant.

Monty (in person): Okay, let's try it out.

Monty reads the script:

Monty (in person): There are three doors here, door number 1, door number 2, and door number 3. Behind one of these doors is a prize, behind the others nothing at all. You can pick one of these doors, and if the prize is behind that door then you have won it. Otherwise, you win nothing at all. So, Contestant, which door will it be?

Producer (as contestant): I'll pick door number 1.

Monty: Door number 1? Now, before I open it, I'm going to show you something. Door number 2...

Producer (aside to Monty): This is where they'll open door number 2 and reveal that there's nothing behind it. Sorry, go on.

Monty: ... is empty. Now, I'll offer you a deal. You could get what's behind the door that you've chosen, door number 1, or you could get what's behind door number 3. Would you rather stick with door number 1, or would you rather get both door number 2 and door number 3?

Producer (as contestant): I, I think I'll stick with my gut on this one. Door number 1.

Producer (to Monty): What do you think? We'd like to pose this one question to the contestant on every show, instead of going through the drawn-out exchange.

Monty: I like it. It's a lot snappier than what we've been doing. It works out the same, but it gets right to the point. It gets there so fast that it may be hard to follow the logic of it, but I like that too. It really puts the contestant on the spot. This is the kind of game show dilemma that I'd like to be associated with.

End Scene.