> If you don't allow for those possibilities, then the host really isn't making their choice with equal probability.
I don't understand your point. The host can make the choice with equal probability and all those possibilities were initially allowed. Once that a choice was made some possibilities are obviously no longer allowed.
> The 50/50 scenario doesn't make sense.
This is the original problem: https://web.archive.org/web/20130121183432/http://marilynvos...
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. [It's implicit that his knowledge is used to pick a goat with certainty.] He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?
This is one possible 50/50 scenario:
You pick a door, say #1, and the host picks a number between 1 and 3 from a hat, say #3, and opens that door, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?
The answer in the first variant is "yes, switching from #1 to #2 doubles my probability of winning". The answer in the second variant is "it doesn't matter, the probability of winning is the same".
The scenarios that didn't happen are irrelevant. We are not been asked to commit to a strategy in advance. We are asked if we want to switch from door #1 to door #2 after we learn that door #3 was not the good one. That depends only the scenarios that still remain possible and their probabilities.
A much simpler problem to stress that the question is whether we want to change conditional on what we know - not all the things that were possible a priory:
You're given the choice between two evelopes. One contains $1 and the other $10. You open an envelope and find $1. Is it to your advantage to switch to the other envelope?
The answer is obviously yes. If you found $10 the answer would obviously be no. But if you have to commit to switching or not before you see what was your first choice the answer would be "it doesn't change anything".
Your answer to the question is conditional on what may have happened. You don't have to "allow for other possibilities" that could have happened but you know already that didn’t.