[jjnoakes]

I disagree. Every person I've ever talked to who struggles with this problem is confused over the probabilities, even after we discuss the situation in great detail. They perfectly understand the game and the rules.

You seem to be finding flaws where there are none. You are suggesting there is subtle unspoken trickery hidden in the problem (just as in my example) but nothing about the problem suggests that should be the case.

What if the host lets you choose to switch every game but if you make the right choice he says you are wrong without opening the door and the game is just over? What if he says the door you chose is eliminated and the remaining door is your prize if you chose correctly? What if you win and then he says you have to play best 2 out of 3 to really win, and if you win again he says 3 out of 5, and keeps moving the goalposts until you lose?

Suggesting the problem is flawed because you can imagine up fringe scenarios that are not explicitly excluded does not seem like a useful criticism.

[Myto]

There being many of you does not make you any more correct. I'm not making up "fringe scenarios". I have clearly (I think) explained how, given the problem as stated, switching is not necessarily beneficial, and can be harmful. You need to make an additional assumption (that Monty necessarily behaves in a certain way) that is not stated in the problem to get to the "always switch" answer.

By the way, once you make that assumption, those other scenarios you presented are also excluded.

[jjnoakes]

Quantity doesn't affect the correctness of facts but it sure affects questions of perception, and your argument seems to be one of perception.

> By the way, once you make that assumption, those other scenarios you presented are also excluded.

Even more of a reason that "flaw" shouldn't even be considered.

There is no hidden trickery and no reason to assume the game isn't fair.

[Myto]

I'll try one more time. Let's imagine there are two worlds, A and B.

In A, Monty Hall behaves like you think: always opens a goat door, always gives the option to switch.

In B, he behaves like I described: opens a goat door and gives the option to switch, but only when the player has chosen the car door. Otherwise he does not let you switch.

And then we find ourselves in this situation:

> 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 No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

At this point, how do you know you are in world A and not B (or some other world)? What in this problem as given here allows you to determine that?

[jjnoakes]

There is nothing in the problem statement that should reasonably lead you to believe there are any tricks at play. The fact that you could come up with some chicanery that the host might pull which isn't explicitly disallowed doesn't make the original problem flawed in any way.

I bet you could take any problem of similar nature from anywhere and scrutinize hard enough and find some gimmick that lets you claim similar claims about it, but I don't think that's useful or noteworthy.

At the end of the day we all rely on common sense and common assumptions about these kinds of things. It may be true that some don't have the same shared experience to draw upon and lead them to the same understanding as others, but that doesn't make the problem flawed. It just means people understand things differently.

If a significant number of people brought up this issue then my opinion might change, but as I said, this is the only time I've heard of this particular complaint (and I've been enjoying posing this problem to people for decades now), and that means, in my opinion, there is no grounds to claim your misunderstanding as some objective flaw in the wording or presentation of the problem.

[Myto]

You reject my argument without being able to point to any flaw in it. Because you and a lot of other people do not accept or have not thought of it. I can't really help with that.

If it's about popularity instead of logical argument, of course I'm not the only one who thinks this. There was a really good blog post that laid it out but I can't find it currently. Instead, here's a scientific paper I found just now, the introduction contains the same argument I'm making. And it's far from the only place which agrees.

https://link.springer.com/article/10.1186/2195-5468-2-2

> The problem posed in this way may lead to a lot of controversy, mainly because we do not know whether the behavior of the host had anything to do with your first choice or not.

> Perhaps the host would open a door with a goat only when your first choice was right. In this case, it was not a good choice to change doors.

EDIT: Also several people have pointed this same thing out elsewhere in this thread.

[jjnoakes]

I pointed out several reasons why I think your argument is flawed.

[AnimalMuppet]

In the real world, you know by watching the show. Monte Hall always offers to switch, so that you know that you're in world A.

If it was the first time the show ever aired, you could be in a situation where you didn't know whether it was A or B. But in the problem as normally posed, you have time to watch the show for months to years, and you know how Monty behaves.

[Myto]

I've never watched the show, but I have read that that is not the case; sometimes he gave the option to switch, and sometimes not. Even if that were the case, it is not stated in the problem and therefore you cannot assume he behaves like he might in the real world. Also even if he had allowed the switch every previous time, it still does not logically mean that he does that the one time you are playing.

And once more, I'm talking about the problem as given, not some other problem. It is a self contained math / logic problem.