The thing you might be overlooking is that Monty does not open a door at random, he opens a door guaranteed to contain a goat. When I first heard this problem, I didn’t get it until that was explicitly pointed out to me.
If Monty opens a door at random (and the door could contain a car), then there is no causal link and therefore the probability would be as you describe.
I’d like to meet some LW people. Couple suggestions in/around Harvard Sq: http://www.cambridgecommonrestaurant.com/ http://www.grendelsden.com/ http://www.johnharvards.com