How to... win a car instead of a goat

My high school maths teacher (Dr V! I wonder what he's up to these days...) went through this probability question with us in a class one afternoon. I still love it, because it's totally logical but for some reason seems illogical when you first hear it. It's one of those things where it either makes you go "ugh, THIS is why I hate maths!" or, like me, you go, "ah, THIS is why I love maths!" So simple if you look at it one way, but totally confusing if you're thinking about it a different way.

The Game Show Problem

So, you're a contestant on a game show and there are 3 closed doors.
Behind one door is a brand new car. The other two doors have a goat behind each one.

The game show host asks you to pick the door you think has the car. He then opens one of the other two doors, revealing a goat.



After opening the door, the host asks if you would like to switch your choice of door to the remaining closed door.



Should you switch your choice to the remaining closed door??

Well, instinct says it really doesn't matter because it's 50/50 at this point, right? There are two closed doors. One has a goat, one has a car.

But no.

You should actually change your choice to the other closed door. You will DOUBLE your chances of winning the car if you do! Weird, huh?

I remember sitting in class and we all looked at our teacher like "um... that's stoooopid, it's 50/50 chance at this point, so why change? duuur..." (because, you know, we as teenagers knew everything about everything... except probability, it turns out...)

But think about this -
Okay, so originally you have a 2/3 chance of choosing the goat and 1/3 chance of choosing the car. So, you have double the chance of choosing the goat as you do the car, right?

Then if one of the goat doors has been revealed and thus that option has been removed from the choices, that means that since your original choice was twice as likely to be a goat as a car then it's twice as likely that the other door is the car.

So switch!!!

If that doesn't make sense still, think about it for a while and it will come together. One of the best pieces of advice my maths lecturer at uni gave to me when I just could not understand a particular theory was that if I look at it long enough, even if I'm just kind of staring at it, it will start to come together eventually. This works.

Xxj