Whether the host opens a door with a goat or a car determines whether the game can still be played at all. How that door is selected does not factor into whether the contestant should switch or not in any way.
I think the source of the paradox lies somewhere in the biases (the Endowment effect or Status Quo bias) as discussed in the Wikipedia article about the problem - not with how the question was stated and especially not with the part about how the host selects a door.
Ah, you are wrong (and this is often confusing to people).
If you model this game where the host chooses randomly, and might open the car door (when that hapoens the player instantly loses) , then there is no point switching.
I think the fact this does matter is the source of many people's confusion.
Why not? Your chances of choosing the correct door were 1 in 3 from the start. That doesn't change with the fact that the host opens the doors at random.
Sorry for the late reply, but if you look at the original article on GitHub I explore that scenario in the code. The answer is that if Monty randomly opens a door then the contestant automatically loses 1/3 of the time before given a chance to switch.
That's what the GP said. In reality, the host doesn't open the door at random. They know which door hides the car and avoid opening it. That's why switching the doors doubles your chances of finding it (because the host has already eliminated one of the goat doors).
Your chance was 1 in 3 yes, but if the host can randomly open the car door, there is a chance of 1 in 3 of that happening, leaving only 1 in 3 chwnce of switching being legal and useful.
I don't think I've ever seen this rule in any variants.
Since we're making up rules: Could the player not still have the option to switch, and thus win the game?
I'm not convinced that I am wrong after your explanation.
Either way, my point is that focusing on the problem statement as the "main thing that seems to cause the paradox" allows the reader to stop there and disregard a far more interesting discussion about biases.
How is that rule made up? There's one car, if that door's been opened it's not left for the player to open. Obviously you can't win by switching to the location you know it's in because you've seen it, that's not a game.
I think the source of the paradox lies somewhere in the biases (the Endowment effect or Status Quo bias) as discussed in the Wikipedia article about the problem - not with how the question was stated and especially not with the part about how the host selects a door.