[Myto]

Pretty much every description of the Monty Hall problem has the same flaw, and it is here also. The problem as given does not describe the general rules by which Monty operates. It describes only a single round of playing the game.

Thefefore, Monty could be using the strategy of "if the player chose the car door, open a goat door and give the option to switch. Otherwise don't give the option to switch and the player wins the goat." In that case switching is a losing strategy.

[jjnoakes]

That's no more a flaw than failing to explicitly say the car is valuable. I mean what if the car is a matchbox toy and the goats are worth more? That's not explicit either.

Sometimes you have to use common sense, and I think every instance of the monty hall problem I've seen was sufficiently explicit (without being absurd), and the confusion was always around the math and probability and never around semantics or trickery.

[Myto]

Obviously the goal of the problem is to get the car. Pretending that the argument I have given is like making up something about a toy car just does not do anything.

The fact is that the argument I have presented demonstrates that the problem as given is flawed and does not have a unique answer. Most people don't understand this and substitute the correct version of the problem in their mind, and then proceed to solve that by arguing about the probabilities. Of course the probabilities are what the problem is "supposed" to be about.

[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?

[spawarotti]

I would not say it is a flaw. I think it is a reasonable implicit assumption that there is only one round, unless explicitly stated otherwise.

[Myto]

It does not matter if there are many rounds or one, what matters is how Monty behaves. And that is given only for the current round, and not as a general rule.