[____a]

My understanding is that in both situations as you described above you should switch. You are still in a new conditional state. Telling you that the host opened another door by chance and it is a goat removes the possibility that he opened the door with the car behind it by chance. You have eliminated one of the options and you have the 2/3 chance of winning by switching.

[throwaway283719]

No. In the first case, the fact that the host shows you a goat has not put you in any new state of the world (the host was going to show you a goat regardless of what happened).

In the second case, you now know that you are not in one of the states of the world where the host accidentally opened a door that had a car behind it. They could have opened a door with a car, but they didn't. That gives you some information - it stands to reason that since they didn't open a door with a car, it was probably a bit more likely that they would pick a door with a goat, which makes it more likely that there are two goats than that there is one car and one goat, which is equivalent to saying that it's a bit more likely that you have already chosen the car.

[____a]

Okay, yes. Had to work it out on paper to convince myself that is works out to 1/2 in the second case. Thank you.