> whether the host is allowed to choose whether to open a door or not
The problem as stated is: "you select door #1. Then, the host, who is well-aware of what’s going on behind the scenes, opens door #3". Why would you think the host has the option of not opening a door? And if you think they could do that, why couldn't the host also do other things, like offer the contestant cash to walk away, or swap what's behind the doors, or add more doors, or open all three doors, or sucker-punch them in the gut?
This is the source of all the confusion from vos Savant's original column. vos Savant got the math right for her version of the problem where the host always opens a door, but that isn't how the show worked. She presented a simplified "spherical frictionless cows" version of the problem.
The actual show did not compel Monty with any rules on opening doors. He had free rein to do whatever he wanted. He could reveal an incorrect guess right away with no opportunity to switch. He could and would offer the contestant cash to switch or not switch. Monty could try anything to trick or bluff or deceive the contestant, all for the sake of producing entertaining television.
Of course, there is no correct mathematical answer at all to a game of bluffing and outwitting. At best you might find a Nash equilibrium in game theory, but even that is hard to pin down. You need to know Monty's tendencies precisely, but even that is hard when his goal isn't to maximize expected value but to produce entertainment.
> The actual show did not compel Monty with any rules on opening doors. He had free rein to do whatever he wanted. He could reveal an incorrect guess right away with no opportunity to switch. He could and would offer the contestant cash to switch or not switch. Monty could try anything to trick or bluff or deceive the contestant, all for the sake of producing entertaining television.
And really, the thing is, none of that matters. Maybe in this version he could reveal an incorrect guess! Maybe in this version he could reveal the door that the contestant picked! That's not what happened in the situation you're asked to analyze. All that matters is that the host picked a door that (a) was not the door the contestant picked, and (b) contains a goat. Nothing beyond that is part of this problem, and nothing beyond that is necessary to solve it. You're given a very specific situation, and asked what the best decision to make is.
> This is the source of all the confusion from vos Savant's original column. vos Savant got the math right for her version of the problem where the host always opens a door, but that isn't how the show worked. She presented a simplified "spherical frictionless cows" version of the problem.
A bunch of PhDs aren't going to have a problem with a precisely specified version of the problem, which is what was asked of her in the original column. (which, incidentally, didn't use the name "Let's Make a Deal" at all).
The problem as stated is: "you select door #1. Then, the host, who is well-aware of what’s going on behind the scenes, opens door #3". Why would you think the host has the option of not opening a door? And if you think they could do that, why couldn't the host also do other things, like offer the contestant cash to walk away, or swap what's behind the doors, or add more doors, or open all three doors, or sucker-punch them in the gut?