Thursday, October 05, 2006

The Monty Hall Problem!!

Kart's rating - 4/5

There used to be a column called Ask Marilyn in a magazine called Parade in America. She had the highest IQ in the world and in the column she answered maths questions sent in by readers. And 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 any door.

Lots of people wrote to the magazine to say that Marilyn vos Savant was wrong, even when she explained very carefully why she was right. Of the letters she got about the problem, 92% said that she was wrong and lots of these were from mathematicians and scientists. Here are some of the things that they said

I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error. -- Robert Sachs, Ph.D., George Mason University

There is enough mathematical illiteracy in this country, and we don't need the world's highest IQ propagating more. Shame! -- Scott Smith, Ph.D., University of Florida

But Marilyn vos Savant was right. And here are 2 ways you can show this.

Firstly you can do it by maths like this

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

The second way you can work it out is by making a picture of all the possible outcomes like this

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.

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.

It also shows that numbers are sometimes very complicated and not very straightforward at all. And that is why I like The Monty Hall Problem.

Source: Curious incident of the dog in the night time, a novel by Mark Haddin.

PS: I really enjoyed reading the book. The author is a chap called Mark Haddon who wrote children’s books and decided to get into the mind of one, but this time an Autistic child. It must have been a daunting & painstaking idea because from the efforts, one can easily make out that he eventually learnt precisely how to go about it. It must have been a really huge effort, because not only did he get into Christopher’s (15 year old autistic child) mind, he also created an engrossing whodunit. It's really funny to find the chapter numbers as 2, 3, 5, 7, 11.... yup, they were prime mumbers.

Bonus: Here are few interesting lines from the book...

1) "It was nice in the police cell. It was almost a perfect cube, 2 meters long by 2 meters wide by 2 meters high. It contained approximately 8 cubic meters of air.

I wondered how I would escape if I was in a story."

2) "I find people confusing. This is for two main reasons.

The first main reason is that people do a lot of talking without using any words. Siobhan says that if you raise one eyebrow it can mean lots of different things. It can mean "I want to do sex with you" and it can also mean "I think that what you just said was very stupid."

The second main reason is that people often talk using metaphors. For instance,

They had a skeleton in the cupboard.
We had a real pig of a day.

The word metaphor means carrying something from one place to another, and it comes from the Greek words µετα (which means from one place to another) and φερειυ (which means to carry), and it is when you describe something by using a word for something that it isn't. This means that the word metaphor is a metaphor.

I think it should be called a lie because a pig is not like a day and people do not have skeletons in their cupboards. "

3) This will not be a funny book. I cannot tell jokes because I do not understand them. Here is a joke, as an example. It is one of Father's.

"His face was drawn but the curtains were real. "

I know why this is meant to be funny. I asked. It is because drawn has three meanings, and they are (1) drawn with a pencil, (2) exhausted, and (3) pulled across a window, and meaning 1 refers to both the face and the curtains, meaning 2 refers only to the face, and meaning 3 refers only to the curtains.
If I try to say the joke to myself, making the word mean the three different things at the same time, it is like hearing three different pieces of music at the same time, which is uncomfortable and confusing and not nice like white noise. It is like three people trying to talk to you at the same time about different things.
And that is why there are no jokes in this book.

Hope, it is enough to make u read the 110 page novel.


Prad said...

Sounds like a gud read...
I shud try it sometime...

kart said...

wYup def it is... but the second half of the book (the train journey to "141/c Chapter Road, London") wasn't all that interesting!!

Iday said...

I've heard abt this book. It is on my To-Read list :)

Iday said...

dude - i read the novel.
it is awesome!!!
i really liked it :)

kart said...

Cool!! The way the novel was written is awesome. Short and sweet.