There are many variants on this game, see the recent book of Rosenhouse. So, 2/3 of the times you will be winning the car. One of the easiest way of seeing this is the following: imagine that you play this game repeatedly, around 1/3 of the times your first choice is right, and then you do not win the car because you change doors in the other 2/3 of the times your first choice is wrong, and changing to the door the host did not open will make you win the car. But if the host always shows you a door with a goat after your first choice, then by changing the first choice you increase your probability of winning a car from 1/3 to 2/3. In this case, it was not a good choice to change doors. Perhaps the host would open a door with a goat only when your first choice was right.
As Gill stated in, this is a problem of mathematical modeling, and the answer is not a probability but a decision, and the decision must be chosen in a setting of uncertainty. The problem posed in this way may lead to a lot of controversy, mainly because we do not know whether the behavior of the host had anything to do with your first choice or not. And then he gives you the possibility to change, allowing you to pick door No. And before opening it, the host who knows where the car is, opens another door, say No. You select one of the doors, say door No. One of them has a car in the backside, and the other ones have goats. Selvin, and it is a nice and controversial problem for introductory courses in probability, statistics, and game theory. The Monty Hall problem appeared first in a letter to the American Statistician of S.
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