> There are two doors with goats and one with a car. You choose one door from the three. The host selects one of the doors with a goat from the remaining two doors, and opens it. Should you switch doors if given the chance?
has a different answer than this one
> There are two doors with goats and one with a car. You choose one door from the three. The host chooses one of the remaining two doors at random and opens it, showing a goat. Should you switch doors if given the chance?
In the first example the hosts choice is either forced (if you chose a goat initially) or doesn't make a difference (if you chose a car).
In the second example, the host could have opened a door with a car because he was choosing the doors at random. Given that he didn't, your estimate of the probability that you have chosen the car should increase - in fact, it should increase to 1/2, and switching doesn't make any difference.
To me, this is the really interesting thing about the Monty Hall problem - the knowledge and intentions of the host are important in the correct solution to the problem! So there are two ways to come up with the answer "it doesn't matter if you switch" -
1. You understand the problem in the first sense above, and your intuition leads you to an incorrect answer.
2. You understand the problem in the second sense above, and you are correct that it doesn't matter whether you switch or not.