[notanzaiiswear]

What exactly do you mean? You mean because it could be the casino scam you describe elsewhere, that they only open the other door if you picked the right door?

Then I think the confusion might stem from the concept of probability. I think what is meant here is probability from the point of view of the deciding person (given their knowledge). That does not have to be the real probability at all.

I don't have a good example and I am not currently well read on it, but I remember from some Bayesian book the example of the lawn being wet. So either it has rained, or the lawn sprinkler had been on. And you can reason about the probabilities. But of course either the one or the other happened (in a simplified logic world). There are no probabilities about it. The probabilities are only the estimates of the person trying to infer if it has rained or not.

OK so strictly speaking, you could create zillions of variants of the Monty Hall Problem from the scarce description. Maybe the host looked at the player funny before the show, so the player raises his estimates for the likelihood of the host being a crook (as in your casino game). Or any number of other things could affect their reasoning.

So it would be merely convention of the mathematical puzzle to assume the simplest case of everything else being random. And then of course there will be puzzles that trick you exactly with that assumption, to teach you that you shouldn't make too many assumptions. OK.