You have removed a constraint from the original problem. What could have happened does not matter since we are being asked about what did happen.
Imagine two people playing a game of chess. Various moves are played. Then you are asked a question about the state of the board. The answer depends on the state of the board. It does not depend on the set of all states the board could have taken had one player made different choices.
Oh yes, the problem text specifies that in this particular instance the host opens a door and offers a switch. It does not specify that the host does this every time, which is the constraint in question.
It's typically considered unnecessary to specify that, because it comes from a game show where he always reveals one wrong door. Monty Hall was the first host of the show.
> because it comes from a game show where he always reveals one wrong door.
Nope!
> Was Mr. Hall cheating? Not according to the rules of the show, because he did have the option of not offering the switch, and he usually did not offer it.
This is about the only proper counter I've read so far. Why did it take so long to post?
This does indeed change the whole problem. I would argue that the problem as stated in vos Savant's column is different (and she says as much later on, "all other variations are different problems"), but I admit this makes me lose the supporting argument I've been using of "...and this is how Monty's game show worked". Point conceded.
I would also argue most people who objected to vos Savant's solution weren't considering Monty's strategy at all. They were objecting to the basic probabilities of the problem as stated by vos Savant, merely because they are counterintuitive (which can be summed up as "if you switch, you're betting you got the first guess of 1/3 wrong"), and everything else is an a posteriori rationalization.
> I posted exactly the same thing in a reply to you way earlier in the thread [1]. It didn't take so long to post, you just weren't paying attention.
I'm not asking why YOU took so long to respond or finding fault in your reasoning abilities, I'm saying there's been a lot of arguing in general in this sub-topic, and few people mentioned this fact -- which is the only relevant fact for challenging vos Savant's formulation of the problem (which matters because it's what sparked all this fuss).
> It didn't take so long to post, you just weren't paying attention.
This is the most dismissive possible thing to say, especially in response to a comment of mine where I'm conceding a point. I missed ONE other particular comment of yours, hence "I wasn't paying attention"? Wow. Sorry for not following your every response to everything.
> [...] the discussion was about Persi Diaconis.
I don't know nor care who Diaconis is, I just care about whether the Monty Hall problem truly was underspecified or not. This is about the Monty Hall problem, not about some person.
Ah, but probability is all about hypothetical instances, and how the host makes his decisions—or if he’s allowed to make a decision at all—is a key consideration in the calculation of the probability. If we don’t know how the host decides whether or not to offer a switch then we can’t calculate a probability and can’t decide which choice is better.
I see your point. You are arguing that the fact that the host did this could convey additional information that would affect the distribution. This criticism still does not seem valid to me because this argument can be used to alter the correct answer to a large number of problems.
Consider the question of whether John Doe did well on his mathematics examination. This would seem like a straightforward thing depending on the questions and his answers. We can assume they are provided as part of the problem statement. We could also assume that a definition for “did well” is included. We could then consider a situation where under chaos theory, his act of taking the examination caused a hurricane that destroyed his answer sheet before it had been graded. This situation was not mentioned as either a possibility or non-possibility. However, we had the insight to consider it. Thus, we can say we don’t know if he did well on his mathematics examination, even though there is a straightforward answer.
Another possibility is that game show could have rigged things without telling us, with a 90% chance of the prize is behind behind door #1, a 9% chance that the prize is behind door #2, and a 1% chance that the prize is behind door #3. Which door was the initial choice would then decide whether the player should change the choice, rather than anything the host does. However, this was not told to us, but to avoid saying that choosing the other door is always the answer, we decide to question the uniformity of the probability distribution, despite there being no reason to think it is non-uniform. Thus, assuming that the game show might have altered the probability distribution, we can say not only that the host’s intent does not matter, but we don’t know the answer to the question.
To be clear, my counterpoint is that these considerations produce different problems and thus are not relevant.
You might live in a world where the host doesn't want to give you the car, and only opens a door and offers you the option of switching if your first choice was the door with the car behind it. In that world, you shouldn't switch. I don't think this form of the problem statement gives you any reason to believe that you aren't in that world.
> 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, I am discussing a different problem, and I don't think the original problem formulation gives enough information to distinguish between the 2 problems.
The answer can add assumptions, which is fine. I'm not passing judgement on Marilyn vos Savant. I do object to claims that the problem statement is sufficient to have a single answer, and based on that, I'd object to claims that somebody in that situation would be wrong not to switch doors. I would object on exactly the same grounds to anyone who tells you "you're wrong, there's a 50% chance of getting a car" (I might object further, on the grounds that the most obvious interpretation which gives that answer is inconsistent with this form of the problem statement).
If you're discussing a different problem, then it's not the Monty Hall Problem, which we're discussing here.
It's a probabilities logic puzzle, it's not about psychological tricks. Anything of that sort is an extraneous ad hoc hypothesis that you're introducing.
The point is whether, upon the reveal of a goat, you should switch or stick to your original choice. Nothing else matters. What Monty had for breakfast doesn't matter. Whether he likes you or not doesn't matter.
Imagine two people playing a game of chess. Various moves are played. Then you are asked a question about the state of the board. The answer depends on the state of the board. It does not depend on the set of all states the board could have taken had one player made different choices.