For me the hangup was always the hidden rule: host won’t open a door with a car. That is unstated and remains unstated even in modern discussions of the problem (see Pinker’s “Rationality”). Once explicitly states the outcome becomes intuitive.
Yes, in many formulations the unstated assumptions that the host (a) will always open a door after your initial pick, and (b) that it's always one without a car behind it. Making the assumptions explicitly makes the solution and intuition much simpler.
If your initial choice was a car, the host can open any of the remaining doors, but if your initial choice was a goat this forces the host to reveal extra information to you (namely which of the remaining doors contains the car). Since your initial probability of picking a goat was 2/3, there is 2/3 probability that the host will reveal the prize door for you.
This is why the puzzle is only loosely based on a TV show. No real TV or other iterated games will work like this, since the optimal strategy is pretty simple. In a real TV show, the host would mix up his strategy (never revealing the car door, but only occasionally opening a door after the candidates choice). In that case it's not possible to work out an optimal strategy without additional assumptions or clues wrt the host behavior. E.g. he might be biased to open a remaining door with higher probability when the initial choice was correct, to increase suspension for the viewers, in which the dominant strategy is actually to not switch. But in a real TV show or iterated game, the host behavior is likely not deterministic.
I do not think that when it's explicitly stated it becomes intuitive for everyone and that adds a further wrinkle. Many people will still get it wrong, even when the problem is stated correctly.
However, if the rule is not explicitly stated, how can the player know that the rule exists? Perhaps "Monty" is evil and will not always open a door, "evil Monty" will only open a door when he knows you've chosen correctly.
IOW, without that rule explicitly stated, the answer "Switch" is simply incorrect. Without that rule, the answer is "I don't have enough information to know."
In fact, the Wikipedia Monty Hall article discusses pretty much any aspect of the problem that anyone has ever brought up in any Monty Hall forum thread or blog post.
em500|3 years ago
If your initial choice was a car, the host can open any of the remaining doors, but if your initial choice was a goat this forces the host to reveal extra information to you (namely which of the remaining doors contains the car). Since your initial probability of picking a goat was 2/3, there is 2/3 probability that the host will reveal the prize door for you.
This is why the puzzle is only loosely based on a TV show. No real TV or other iterated games will work like this, since the optimal strategy is pretty simple. In a real TV show, the host would mix up his strategy (never revealing the car door, but only occasionally opening a door after the candidates choice). In that case it's not possible to work out an optimal strategy without additional assumptions or clues wrt the host behavior. E.g. he might be biased to open a remaining door with higher probability when the initial choice was correct, to increase suspension for the viewers, in which the dominant strategy is actually to not switch. But in a real TV show or iterated game, the host behavior is likely not deterministic.
nmdeadhead|3 years ago
However, if the rule is not explicitly stated, how can the player know that the rule exists? Perhaps "Monty" is evil and will not always open a door, "evil Monty" will only open a door when he knows you've chosen correctly.
IOW, without that rule explicitly stated, the answer "Switch" is simply incorrect. Without that rule, the answer is "I don't have enough information to know."
em500|3 years ago
In fact, the Wikipedia Monty Hall article discusses pretty much any aspect of the problem that anyone has ever brought up in any Monty Hall forum thread or blog post.