Let's make a deal...
Category: Statistics
Published: 01/14/2009 08:23 a.m.
In the movie 21, Kevin Spacey plays an instructor at MIT. In one of his lectures, he proposes a common world situation that relates to math. Here is the situation...
"You are on Let's Make a Deal. There are three doors (1,2, and 3). One has a car behind it and the other two have goats. Monty asks you to pick a door. Then, he opens another door and shows you a goat. Then he asks if you want to switch your door to the remaining unopened door."
Do you take the switch?
Most people have a tendency to not trust the option, so they keep their door. Other people (who think they are smart) say that there are two doors left, so they have a 50/50 chance with the door they originally picked (relying on emotion), so they will keep it. Both of these people are wrong, and 2/3 of them will end up with a goat.
The really smart people will take the switch. They first are able to separate emotion from logic (which is the point in the movie), but the second part is they understand the math. The simplified version is that by showing you a goat behind a door, that 33% is absorbed into the remaining door. But, most people don't believe this, so here is another explanation with some examples.
Let's say the car is behind door 1. There are three original options and we will look all of them in both cases. Let's say you choose door 1. Door 2 is opened (with a goat) and you are offered a chance to swap to Door 3. If you don't take the swap, you win. If you take the swap you lose. Score: Swap 0/1, No-Swap 1/1.
Now lets say you choose Door 2 at the beginning. Door 1 still has the car, so Monty is force to show the goat behind Door 3. You now have a chance to swap to Door 1. If you swap, you win, and if you don't, you get a goat. Score: Swap 1/2, No-Swap 1/2.
Now the final option. You start with Door 3. The car is still behind Door 1. Monty is now forced to open Door 2, the only other door with a goat. You have the option to swap to Door 1. Again, if you swap, you win and if you don't, you don't. Final Score: Swap 2/3, No-Swap 1/3.
The core of this hinges around picking right at first. In the situation that you guess right (which is 33%), you lose on the swap. But, if your original guess is wrong (66%) and you always swap, you always win. This is very counter-intuitive, because from the start, assuming you will always get the option to switch, your real goal is to pick wrong. And since two of three doors are wrong, you will win with a swap 66% of the time.
Perhaps this (ugly) rudimentary table will help.
| Car is behind Door 1 | ||
|---|---|---|
| swap | no-swap | |
| Pick Door 1 | Lose | Win |
| Pick Door 2 | Win | Lose |
| Pick Door 3 | Win | Lose |
| Total | 2 | 1 |