[n4r9]

Thank you for the explanation; I think I see it now.

Here's another way of seeing it which I think may be effectively the same. Suppose that there are 100 doors in a circle. You pick one and then all the doors are opened except for the one you picked and the next door along. If the car hasn't been revealed yet, it seems intuitively right that you have a 50-50 chance of winning whether you stick or switch at that point.