I'm with you. Usually when it comes up in pop culture, it's not mentioned that the host can't open a door with a prize. That's the only reason it's confusing.

And that they can't open the door chosen by the contestant! Also perhaps more subtly, that Monty picks at random when given the option. The original problem prompt is _highly_ underspecified (and makes assumptions that are counter-intuitive to how a game show host might behave).

The problem has posed by von Savant specified the host knew where the prize was. And claims the host reveals a non prize for. It is fairly obvious the host won't reveal the winning door, as it would end the problem. Not sure how the assumptions are fighter intuitive.

Of course Hall himself said maybe I open a door because I want you to switch.

What you're missing here is that you already have enough intuition to translate it into a more obvious statistical problem, but most people come across this as blank slates, and under very different framing.

Even when relayed the right way and the wording isn't necessarily unclear or misleading per se, it is still framed in a particular way and context that pushes you to believe your choice is inconsequential, less about probability and math, and more for psychological games and dramatic effect in the context of a TV show.

In your example, the fact that the presenter opens all 98 doors and not just one is telling. If they only opened 1, and then told you to stick to your original or open one of the remaining 98 doors, that he picks for you, would you change?

This is what you're missing, I think. Most people don't even get to the point where they reframe the "would you change" question to "what are the chances my original choice was correct vs not". Most people get stuck on "He knows something; he clearly demonstrated very theatrically that he knew the car wouldn't be behind door No.2. And now he's asking me to swap my choice for door No.67. Why 67? Is he trying to trick me? Is this a psychological trick to get me to forfeit my door, which may be the right one? Should I trust him? After all, HOW IS MY DOOR ANY DIFFERENT THAN ALL THE OTHERS?"

(the fallacious thinking, obviously, lying in the last part here)

One thing you're missing: "The host of this game, who knows where the prize is, then opens one door without a prize"

Does the host _always_ do this, or does the host have discretion about _whether_or_not_ to do this? This has not been stated in any of the problem descriptions here, and I believe it's highly significant. If the host can choose whether or not to do this, then there is the potential for mindgames. If the host cannot choose whether or not to do this, that must be stated.

You are not missing anything but you have gained 97 doors!

To get the subtlety here, you probably need to model it which is what I did. That's why I'm an IT bod these days, with a Civil Engineering degree and not a mathematician or statistician.

YOU aren't missing anything. But a lot of people can't give over "two doors means fifty fifty"