If we know that the host will always show us a goat behind another door, then yes, we should clearly switch.
If the host typically lets us just open the door, but will show us a goat before we open the door if the show is running too fast and they need to kill time, then we should switch if offered.
If the host typically lets us open the door, but will show us a goat if the show is running too fast, or if the prize budget is running low and we picked the car, then we should switch if we think the previous games went quickly, but not if there were some slow games already.
If the host only shows a goat when the contestent picks the car, then we should never switch.
Many problem statements include that the host always shows a goat; and if it doesn't you can kind of assume it, because it's a well-known problem, but if it's a novel problem and unsaid, then how are you supposed to know? I haven't watched enough Let's Make a Deal to know if they always give a second choice. Reading the NYT article linked elsewhere in the thread, I am reminded that Monty Hall could offer cash to not open doors too, so with the problem as stated and Let's Make a Deal being referenced, I have to assume an antagonistic host, unless provided with more information on their behavior.
As stated, assuming unknown behavior of the host, we can't put a number on the probability of switching.
Also, to address another point you made elsewhere in the thread. In addition to specifying the host behavior, it should also be specified in the problem statement that the car and goat positions were determined randomly, or at least the car was random, and the two goats are considered equal and assigned as convenient.
> So let’s look at it again, remembering that the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. (There’s no way he can always open a losing door by chance!) Anything else is a different question.
https://web.archive.org/web/20130121183432/http://marilynvos...
What you are discussing is a different problem.
The question does not state that. It's an assumption that was made in the answering.
If that's a premise, then yes, always switch.
If you only go by the question, there's not enough information.
Are you are accepting that the host might be someone who only opens a door with a goat when your first choice was the door with a car behind it, and still arguing that you should switch?