RE: Would you switch?
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What I find interesting is:
- If the door eliminated is removed by chance rather than by knowledge of the penny's location, and it doesn't result in the penny being behind it, then there's no advantage to swapping doors. It becomes 50/50 between the two remaining doors.
- So you need to know whether the person eliminating the door did so out of knowledge or out of luck in order to determine the odds of whether swapping is beneficial.
The same is true with 1 million doors:
- If someone removes the 999,998 incorrect doors through knowledge then you can be pretty much certain that swapping will give you the penny.
- But if they somehow manage to remove 999,998 doors through chance without knowledge, then there's a 50/50 chance with the two remaining doors.
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Your scenario cannot happen because the host is not allowed to open the chosen door (otherwise 'stay or switch' doesn't make sense). So if the contestant picks a penny door, the host is forced to open the other penny door.
Sure. In a game show that methodology cannot be applied. The host cannot open a door using chance in case they pick the winner.
But outside of a game show, say as a puzzle in a book, the idea holds that:
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Oh, I see what you are saying now. Your setup is a one-time scenario, and the contestant doesn't know the rules of the host. (e.g. the host rule was to flip the other doors randomly, and it happened to get lucky and not reveal the 1M, or the host in this case just opened the penny door and tells the contestant that it was flipped randomly).
But I think the wording of the original post is clear that "one of the doors with a penny behind it will be opened", and that implies it cannot be done by chance.