I wrote a Medium piece on this problem from a lawyer's perspective. As a lawyer, I had a rule about legal analysis: State all and only the facts that apply, and apply all and only the facts you state. This article would need at least two changes to follow that rule.

First, an unstated fact: The contestant knows that the host must open the door that the host knows hides a goat. We are told that the host knows what's behind the door, but not that he must open another door no matter what. Maybe he only does that if the contestant has chosen the door with the car.

Which leads us to a stated fact that isn't used: You are on a game show. This fact, which is crucial to the practical problem facing the contestant, reinforces the problem that the contestant does not know under what conditions the host opens a door. Maybe, if the contestant has chosen the door with the car, the host does not open a door but offers $1,000? On a game show, that might happen.

I wonder how people would handle the question if it were properly expressed:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, and you know that the host, who knows what’s behind the doors, must now open another door that has a goat. He does so and then says to you, “Do you want to pick the remaining door?” Is it to your advantage to switch your choice?

I am not persuaded by footnote 6. Because this is a game show, the host's operating rules are a necessary fact. To test, restate the problem this way:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say №1, and the host, who knows what’s behind the doors, opens another door, say №3, which has a goat. (Unbeknownst to you, the host only opens a second door if you have chosen the door with the car.) He then says to you, “Do you want to pick door №2?” Is it to your advantage to switch your choice?

Clearly, your best move would be to stand pat. So the issue is game-show mechanics, not probabilities. The contestant must guess at the rule driving the host's conduct, a rule that is beyond probabilistic analysis.

Contrary to the statement in the footnote, the contestant knows that there are a limited range of ways in which the situation may have arisen. And one of those ways, which has a higher probability than most others, is that the contestant is on a game show where the host only opens the second door if the contestant has chosen the door with the car. There is no reason to arbitrarily assign equal weights to all possibilities. As a practical matter, the best choice cannot be computed, at least not in the straightforward way that is often used.