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 number 3, and the host, who knows what's behind the doors, opens another door, say number 1, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?
"It's fifty-fifty because there's only two doors."
"Lucy, I've got some 'splainin to do."
The statement of the problem is crucial for the students to understand why the problem isn't simple probability, choosing between two doors randomly.
It's because Monty didn't choose randomly that you don't. Monty KNOWS where the prize is. This changes things from "Monty chooses one of two doors randomly" which would make yours a 50-50 choice as well. Instead, it's "Monty's knows" and avoids the real door which messes up the probabilities. Realize that Monty never, in all those years and all those shows, ever opened a door and said "Ooops, there's a car, you lose."
Like for all of you, that never helps with my students either so I try the twenty doors version next. Choose door 1. Monty opens all the doors except number 13. Stay or switch to #13?
C O O O O O O O O O O O O X O O O O O O O
Everyone wants to switch. But no one will accept the same reasoning will hold for the 3-door version. "Because we are dealing with only two closed doors. DUH." "It's still 50-50, Curmudgeon."
So I ask the students for a brute force solution. How many different possibilities are there? What are the results? It comes quickly to this:
Here are all the possibilities:
| Door Chosen | Prize Door | Monty Opens | Stay Wins | Switch Wins |
|---|---|---|---|---|
| 1 | 1 | 2/3 | 1 | 0 |
| 1 | 2 | 3 | 0 | 1 |
| 1 | 3 | 2 | 0 | 1 |
| 2 | 1 | 3 | 0 | 1 |
| 2 | 2 | 1/3 | 1 | 0 |
| 2 | 3 | 1 | 0 | 1 |
| 3 | 1 | 2 | 0 | 1 |
| 3 | 2 | 1 | 0 | 1 |
| 3 | 3 | 1/2 | 1 | 0 |
Pretty plain. Switch wins 2/3 of the time.
Posts
I love this problem. It's counter intuitive to students. I usually have a flash version of the game and have the students guess what to do. They try and do it there way with disastrous results :-)
ReplyDelete