Switch doors or not?

[RagnaToad]

It certainly does hold up mathematically speaking. Think of it like this:
You choose a door and have 1/3 chance that it's the car. Meaning that when you switch later one, you have a 1/3 chance to switch from the car to a goat.That means you have the rest of the 3/3 -> 2/3 to switch to the car.

Or with a simple [Number of favourable possibilities] divided by [Total number of possibilities] for switching:

Car -> switching means Goat
Goat A -> switching automatically means Car (other goat is eliminated)
Goat B -> switching automatically means Car (other goat is eliminated)

2 out of 3 results of switching get you the car.

NOTE: Eliminating 1 door BEFORE choosing would obviously alter chances from 1/3 to 1/2 of choosing the right one (assuming you want the car ).