Here's been a bunch of questions on the Monty Hall problem, so I'll assume people know the basics.

This answer helped clarify a few things for me, but talking with some colleagues yesterday, someone brought up the idea that as you increase the number of doors, the probability of winning the car by switching approaches 1. Intuitively, this makes sense, but I also know that infinite sets aren't always intuitive.

Is it really the case that if the number of doors increases to infinity, the probability of winning approaches 1? If not, why not?

EDIT: To clarify, the infinite case would look as follows:

- There are an infinite number of doors. One has a car behind it, the others have goats.
- The host knows which door has the car behind it.
- The contestant selects a door
- All doors except the selected door and an additional one are opened revealing goats.
- The contestant is asked if they would like to switch doors.

The question is: in this case, does P(winning by switching doors) = 1? Or is this even well defined? What changes when the number of doors is large but finite?