[full_monty]

Yes.

If Monty opens the door by chance there are 3 equally likely cases: There's a 1/3 chance you picked the car and Monty shows you a goat, 1/3 chance you picked a goat and Monty shows you the other goat, 1/3 chance you picked a goat and Monty shows you a car. So if Monty shows you a goat you have equal probability of being in one of the first two cases.

If Monty doesn't open the door by chance then he never shows you a car. So 2/3 of the time you picked a goat and Monty shows you the other goat.

[krackers]

Yup, the fact that you only have an advantage in switching when Monty "leaks information" in explicitly choosing _not_ to open a certain door as [1] pointed out, is likely the crux of what makes this unintuitive, since it is a very unusual prior.

[1] https://news.ycombinator.com/item?id=39514463