I think we're disagreeing about how much it's reasonable to assume. I'm happier treating it as a self contained problem (in which case I'd say that the form quoted by CrazyStat is underspecified); but if you're familiar with the TV show it's based on, you can reasonably assume that he always opens a door with a goat and gives you a chance to switch.
My objection is to the claim that "most people get it wrong", if most people are being fed the underspecified problem. I think the gut reaction is not to switch, because in most comparable situations across human experience it would be a mistake (imagine a similar situation at a sketchy-looking carnival game rather than a TV show). They then try to justify that formally and make mistakes in their justification, but the initial reaction not to swap is reasonable unless they've been convinced that Monty Hall always opens a door with a goat and gives a chance to switch.
> My objection is to the claim that "most people get it wrong", if most people are being fed the underspecified problem. I think the gut reaction is not to switch, because in most comparable situations across human experience it would be a mistake (imagine a similar situation at a sketchy-looking carnival game rather than a TV show
This may have a role to play. However there is a long history of people who aren't "going off their gut", including statisticians, getting this wrong with a very high level of confidence. It seems pretty clear that there is more than just an "underspecificity" problem. If you properly specify the problem, you will get similar error levels.
I agree, but I believe the reason for the errors is because people intuitively have a pretty good grasp of the game theory for the situation where someone is trying not to give you something they promised (and it's the sort of thing where IRL you shouldn't believe somebody trying to convince you to change your mind, so it's a useful bias to ignore parts of the problem even when it is fully specified). I believe that the statisticians then try to justify that, and end up making incorrect arguments.
> I agree, but I believe the reason for the errors is because people intuitively have a pretty good grasp of the game theory for the situation where someone is trying not to give you something they promised
Unfortunately this doesn't match reality. The vast majority of people who got the problem wrong when it was first published are not confused about the rules and insist that the chances are even (same chance to get a car switching as not switching). This doesn't match a theory that these people think the host is trying to trick the player in some way.
Additionally, You can reframe the problem and will still see significant error rates.
It contains a clarification that the article omitted from the description:
> So let’s look at it again, remembering that the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. (There’s no way he can always open a losing door by chance!) Anything else is a different question.
Yes, of the host opens a door, it will always be a losing door. Nobody is disputing that.
The part that is underspeified is: does the host always open a door and if not, does the player's choice of a door impact whether the host opens a door?
My objection is to the claim that "most people get it wrong", if most people are being fed the underspecified problem. I think the gut reaction is not to switch, because in most comparable situations across human experience it would be a mistake (imagine a similar situation at a sketchy-looking carnival game rather than a TV show). They then try to justify that formally and make mistakes in their justification, but the initial reaction not to swap is reasonable unless they've been convinced that Monty Hall always opens a door with a goat and gives a chance to switch.