They told me I was wrong, and this was their argument: if I don't change doors, then it's just as if I wouldn't have been given the opportunity to change, so I still only have 1/3 chance of winning. Get it? If I don't change, then it's the same scenario as if Monty Hall would just start opening the doors in just a random order and I wouldn't have had the choice to change.
While essentially that argument holds true, it's this easy to stand against it: "Ok, I didn't change doors when I was offered the chance, but I rethought my choice, and I happened to choose my original door again; obviously that gives me the same chances as if I would've chosen the other door!"
So for months I went on thinking I was right, and all I have to do is just consider changing to the other door, but I don't actually have to change.
Well I was wrong!
Truth is, if you change your choice you have twice as much chance of winning than if you stay (2/3 to 1/3).
If originally you choose the door with the car (1/3 chance), then you lose if you switch.
If you originally choose a door with a goat (2/3 chance), then you win if you switch!
It's that simple!
Imagine it this way: same game, except with a deck of cards (52 cards) and the goal is to choose the ace of spades. Someone chooses a card, and you reveal 50 out of the remaining 51 cards. If you imagine it, then most of the time (51 times out of 52 tries) the person will NOT choose the ace of spades right away, and he/she should definitely switch to the card you left for him/her.
What's amazing about this problem?
-How it seems against my instincts that I will be more likely to win if I switch
-How information changes the whole problem!!! In an amazing and unbelievable extent!!!
The information (not-really) paradox:
Imagine that you're playing this Monty Hall game, you originally chose door A, and Monty Hall, the host, reveals that there is a goat behind door C (so now doors A and B are left). Now you have the chance to switch, but before you do so, another player comes in. Monty Hall tells him that behind one of the doors is a car, behind the other there's a goat. The player's default choice is set to door A.
He SHOULD HAVE a 50/50 (1:1) chance of winning!!! But he doesn't!!! He actually has 1:2 chances!! If he chooses to stay, then he will probably lose, and if he switches, he will probably win!
And the weird thing is, the more doors there were in the beginning, the more distorted his odds become! The history of the two doors isn't irrelevant! If there were 50 doors to start with, and 48 were revealed, then his chances per door are 1/50 and 49/50... and the poor guy just has to make the choice between two doors!
He can not have a 50% chance of winning, because if now we separate the room into two parts with a curtain, and you make a choice and he makes a choice, then door A will win 1/50 times and door B will win 49/50 times!!
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