monty hall problem

i recently caught the show 21, where a bunch of MIT students picked up what they call as card-counting techniques to beat casinos at blackjack. 

the lead was chosen because that he answered the monty hall problem.

simi lai monty hall?

well, its actually a gameshow-based question. if you have 3 doors where there is 1 car and 2 goats behind them, you pick a door out of it. if you get the car behind, BINGO! otherwise, you win the goat. bleah... 

so, you choose door 1. the host opens door 3 for you and there's a goat behind it.

now the host asks you "would you like to switch your door?"

aye or nay? 

i bet you'd say nay. the gameshow host is just trying to confuse you right? you got 50-50 chances of getting it right?

actually the answer is aye. change the door. this is the best explanation that i found so far. you'd prolly need a bit of statistical or mathematical head to grasp this. i took many a night to grasp the concept behind it. 

3 doors - Red, Green and Blue. 

P is for probablity. 

1. "Let us call the situation that the prize is behind a given door A_r, A_g, and A_b. To start with, , and to make things simpler we shall assume that we have already picked the red door."  That means the probablity of getting the prize behind any door is a third.
2. Let us call B "the presenter opens the blue door". Without any prior knowledge, we would assign this a probability of 50%. 
3. In the situation where the prize is behind the red door, the host is free to pick between the green or the blue door at random. Thus, P(B | A_r) = 1 / 2
4. In the situation where the prize is behind the green door, the host must pick the blue door. Thus, P(B | A_g) = 1
5. In the situation where the prize is behind the blue door, the host must pick the green door. Thus, P(B | A_b) = 0

Thus,

So, we should always choose the green door.

Note how this depends on the value of P(B). Another way of looking at the apparent inconsistency is that, when you chose the first door, you had a 1/3 chance of being right. When the second door was removed from the list of possibilities, this left the last door with a 2/3 chance of being right.