Monte Hall Logic and Probability

I've read about the Monte Hall Problem in a few different places.

Basically, the situation is that you are on a game show, and there are 3 doors in front of you. There is a car behind one door, and goats behind the other two. If you pick the door with the car, you win the car!

So, you pick one of the doors. Now, at this stage in the game, the host always opens one of the two remaining doors, which always has a goat behind it.

Now, there are only two doors to choose from. The host now gives you the chance to either stay with your choice, or switch to the other door.

What should you do?

The answer, which generates some controversy, is to always switch, because there is a 2/3 chance that the car is behind the other door. A lot of people think the odds are 50-50, but this isn't so.

When you first picked your door, there was a 2/3 chance that the car was not behind the door you picked. For convenience, let's say that you picked door A, and the other two doors are B and C. So, there is a 2/3 chance that the car is behind B or C.

Now, the host opens door B. The key is that, when the host chooses a door to open, he always chooses a door with a goat behind it. That means the host knows where the car is, so his choice is not random. This is why there is still a 1/3 chance that the car is behind door A and a 2/3 chance that the car is behind door C.

I read somewhere that this problem is easier to understand if we consider more doors. So, let us say that there are 100 doors. The car is behind one of the doors. You pick your door, and then there is a 99/100 chance that the door is behind one of the other 99 doors.

Now, the host opens 98 of the doors, revealing goats. So, now you are down to 2 doors: the one you picked and the one that the host did not open. Now, it should be a little easier to see that there is a 99/100 chance that the car is behind the second door, and that you should switch.

In both cases, the second door survived a "test" of the host unveiling goats, while your initial choice was never subjected to a test.