The Monty Hall Problem

The Curious Incident of The Dog in The Night-Time by Mark Haddon has been an interesting read thus far
Socks> I bought it at Kinokuniya, along with a MNG jacket, MNG skirt and a ZARA top
I'm highly recommending this book to avid mathematicians like Socks
who could probably enlighten me on some of the mathematical problems
the protagonist Christopher John Francis Boone is so oft in contemplation,
like the one i'm going to raise here, which i find highly enlightening
with regard to my poor performances at Multiple Choice Question tests.

Before i reveal the problem, some prior knowledge on the context:

There used to be a column called Ask Marilyn in a magazine called Parade, in America. And this column was written by Marilyn vos Savant and in the magazine it said that she has the highest IQ in the world in the Guinness Book of World Records Hall of Fame. And in the column she answered maths questions sent in by readers.

In September 1990 this question was sent in by Craig F. Whitaker of Columbia, Maryland.

You are on a game show on television. On this game show the idea is to win a car as a prize. The game show host shows you three doors. He says that there is a car behind one of the doors and there are goats behind the other two doors. He asks you to pick a door. You pick a door but the door is not opened. Then the game show host opens one of the doors you didn't pick to show a goat (because he knows what is behind the doors). Then he says that you have one final chance to change your mind before the doors are opened and you get a car or a goat. So he asks you if you want to change your mind and pick the other unopened door instead. What should you do?

Marilyn vos Savant said that you should always change and pick the final door because the chances are 2 in 3 that there will be a car behind that door.

But if you use your intuition you think that chance is 50:50 because you think there is an equal chance that the car is behind the door.

Lots of people (mathematicians, scientists, Ph.D holders) wrote to the magazine to say that Marilyn vos Savant was wrong, even when she explained very carefully why she was right."

So Christopher cleverly works it out to show that Marilyn vos Savant was right. He concluded that there are 2 ways to show this. (Now the confusing bit... stop if it's boring you. Only socks will get to this point i think).

Solution 1:

Let the doors be called X, Y and Z.
Let Cx be the event that the car is behind door X and so on.
Let Hx be the event that the host opens door X and so on.
Supposing that you choose door X, the possibility that you win a car if you then switch your choice is given by the following formula:

P(Hz^Cy) + P(Hy^Cz)
= P(Cy).P(Hz|Cy) + P(Cz).P(Hy|Cz)
= (1/3 .1) + (1/3 . 1) = 2/3

(Yah ... Blank face. I can't make any sense out of this too. P stands for permutation i figure. *smug*)

Solution 2:

So, if you change, 2 times out of 3 you get a car. And if you stick, you only get a car 1 time out of 3. (This explanation's more layman and easier to digest, tho' still a little confusing).

And this shows that intuition can sometimes get things wrong. And intuition is what people use in life to make decisions. But logic can help you work out the right answer.

Whoa! *enlightenment*
No wonder i always do badly for MCQ.

P.S. Socks you should get the book.