> I think you're reading the description incorrectly. It is not meant to be a specific instance that may happen differently for other instances. "You" is an abstraction over all players of the game show, there's an implicit "forall you:" in front of the sentence.

But the problem described a specific instance. You are reading it wrong. Right now you are just digging deeper into a hole.

If the problem actually was told as you describe then you would be right, but it isn't. It just describes a single instance, you are at the show, the host opens a door, what do you do?