That's not right, and it's one of the more confusing parts of the problem in my opinion. I can make sense of it by drawing out the probability trees and see th

That's not right, and it's one of the more confusing parts of the problem in my opinion. I can make sense of it by drawing out the probability trees and see that it's 1/2 rather than 2/3 in this case, but it actually sticks with me if I think more like so: My prior is that there's a 1/3 chance I picked the car; that's the world where the host can't reveal anything other than a goat. In the scenario where the host then reveals a goat intentionally, my prior doesn't change. The host can always reveal a goat. This gives me no information. Therefore, it's still 1/3 that I picked the car, so the remaining door must be 2/3, so I switch and get a 2/3 chance. But in the scenario where the host reveals a door at random, and it happens to be a goat, it's time to update my priors. Since he didn't accidentally reveal the prize, the probability that I'm in the world where he couldn't accidentally reveal the prize is increased relative to the probability that I'm in the world where he could have.