You should take the host up on his offer to switch doors. (Most people think either that you shouldn’t switch or else that it doesn’t matter.) By switching, you double your chance of getting the car.
When you make your initial pick of one of the three doors, you have a 1 in 3 chance of being right. That means that there’s a 2 in 3 chance that the car is somewhere else. When the host shows you a door with a goat behind it, that doesn’t change the chance that you were right on your initial pick. (Since there are two goats, he can always find a door with a goat behind it that you haven’t picked.) You still only have a 1 in 3 chance of having been right. The bit of information he’s given you only tells you that there’s no chance the car is behind that door. Since there’s a 2 in 3 chance that the car is somewhere other than behind the door you picked, switching doors increases your chance of getting the car from 1 in 3 to 2 in 3.
[After posting that on a mailing list, I got this reply. Following is (part of) my answer.]
Bill wrote:
> This is a simple odds exercise and I’m not a neophyte Rob. It’s been years,
> but I took every class UNLV offered in gaming theory and also assisted in
> teaching several 100 and 200 level courses (where simple probabilities like
> this problem were well covered).
>
> If you consider my explanation [in separate correspondence, not quoted here]
> to be defective then I’d ask that you point
> out the specific flaw. To be honest, I don’t think you can identify an
> error
It’s interesting that you’ve had that much background and still got it wrong.
I’ll offer two different approaches to showing what the error is. The basic answer is that it’s just not true, as you said, that “if a door is then eliminated the odds change.” Nor is it true that, after a goat is revealed, “zero information is known about the two remaining doors.”
1. Here’s a different way of putting the point that I already made. Think about the choice as part of a series. Two times out of three, you will have picked a door with a goat behind it first. When the host shows you a goat, both goats are accounted for, so switching gives you a car. Only one time out of three will you have picked a car first, so only one time out of three will switching give you a goat.
The fact that you had a one in three chance initially “flows through” because once you pick a door with a goat behind it – which you will do two times out of three – the host is constrained as to what door he can open. There will only be one door he can open to show you a goat.
2. If you don’t find yourself convinced by that, run a simulation of a series of significant length (20 should be plenty). Get a friend with a notepad to try out (with some randomizing procedure) different arrangements of goats and car, and make guesses before he tells you where a goat is. Then, stick with your guess and see how it turns out. Or do it the other way, making a practice of switching. You’ll find out that I’m right.
