The Monty Hall Problem

Familiar with The Monty Hall Problem?

I first came across it in the Curious Incident of the Dog in the Nighttime (which I just happened to mention yesterday, haha.)

When I first read about it, I was extremely confused and couldn't understand.
I mean, I would have gotten it if I had continued to fuss over the problem, but I was anxious to get on with the story and dismissed it, planning to mull it over later.

But you know me la.
I forgot it as soon as I put the book down.

Today, I just read Xiaxue's latest update and she mentioned The Monty Hall Problem.
After reading, I only got the barest gist of the solution - so I decided to Wikipedia it.

And I find that I totally understand. -_-
I had this notion that it was something to do with the initial chances of picking the correct door and it does! =)

The Monty Hall Problem:

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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat.

He then says to you, "Do you want to pick door No. 2?"

Is it to your advantage to switch your choice?

The answer is yes.
Switching doors would give you a 2/3 chance of winning the car.

Many people, including lots of mathematics professors have argued that it is ridiculous and that since there are 2 doors left, one has to have the goat and the other the car; therefore your probability of winning is 1/2 - switching would benefit you in no way at all.

However, this is wrong.

This is becase the initial probability of the contestant picking the CAR in the first place was not taken into account.

Let's say we have 3 doors.
1 car, 2 goats.

First door you pick -
Probability of getting the car is 1/3.
Probability of getting a goat is 2/3.

Let's say that the car is behind Door 1.
Door 2 and Door 3 have goats behind them.

Note that the host can ONLY reveal goats, and not the car.

Scenario 1:

You pick Door 1 - the car.
Host opens a door with a goat behind it.
Switching doors: You get a goat.

Scenario 2:

You pick Door 2 - a goat.
Host opens a door with a goat behind it.
Switching doors: You get the car.

Scenario 3:

You pick Door 3 - a goat.
Host opens a door with a goat behind it.
Switching doors: You get the car.

Therefore, the probability of winning the car AFTER switching is 2/3.
=)

How interesting!
Maths rawks.


Song of the Day:

"Sober - Muse"

^_^ Faham?
If not, you can try reading the original Wikipedia article without me cutting the chiong hei crap here and Xiaxue's blog post here (near the end of the post.)