Monday, November 03, 2008

I learnt something new..

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 other two, a goat each. You pick a door, say Door 1, and the host, who knows what are behind the doors, opens another door which has a goat (purposely), say Door 3. He then says to you, "Do you want to switch to Door 2?" Is it to your advantage to switch your choice?

Many people will assume that there is no difference, since now there are only two unknowns, so it's a 50-50 chance of getting the car. Well, if you think that way too, then you are just another one of the many people who got it wrong. =) But well, seems that even matheticians and even Nobel physicists can get it wrong too, so who can blame us?

The scenarios that may occur:

- The player originally picked the door hiding the car. The game host must open one of the two remaining doors randomly. (therefore chances are 1/2 and 1/2)
- The car is behind Door 2 and the host must open Door 3.
- The car is behind Door 3 and the host must open Door 2.

If it's not enough, how bout the below mapping? =) This helped me to comprehend the situation better but it essentially means the same as the above diagram. Switching still yields a 2/3 chance of winning the car.

So, we should always switch! If you don't believe it, use a deck of cards to prove it. Ace for the car, and any other two cards (say, 2 Kings) for the goats. =) Try experimenting it many many times. You will pick the Ace on average 2 out of 3 times if you choose to switch your initial choice. A simpler explanation I can think of is that: Initially, you stand a 1/3 chance of picking the Ace but 2/3 chance of picking a King. When a King is revealed (not your choice of card), you are left with only 2 cards, an Ace and a King. So by switching, it means that you hope you did not choose the Ace initially and thus switching will get you the Ace! Meaning that you believe you picked the King initially, which is a 2/3 chance and thus 66.7%. Get it?

Hee. Fascinating, right? I think so ley. It's like the Prisoners' Dilemna. Hee. Always logical and there's always a way to explain it based on statistics or probability. Things in life ain't always so logical, but well, I guess I am still very much intrigued.

Tata. So that's all. Just wanna share my new knowledge. Woohoo. Swedish exam tonight. Hope both me and jw will pass the course. Ganbatte ne! (woops, that's japanese, but well. I dunno the equivalent in swedish. Hehe. =p)