It is the same game. The original problem reads:

> You are on a game show. There are three doors, one has a prize behind them. You pick a door. The host who knows whats behind the doors opens one of the doors and reveals that it was the wrong door. Should you switch?

This description doesn't contradict my scenario at all. it doesn't say "The host will always reveals a dud no matter what happens".

Now, your intuition tells you that my scenario feels wrong since the host doesn't always do the same thing every single time, but that is how math and statistics works. We only know that the host opened this door this one time, we aren't given information about how the host acts in general. My interpretation is therefore just as valid.

Of course, for the real game the host always opened a dud door. But the original problem description didn't say that, it just described a single instance. In a single instance there isn't enough information to tell. And no, if you assume that the reader must know how the original game show worked then the problem isn't a properly defined problem, and the answer could be anything. That is how math works.