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> Also, Monty always opens a door with a goat. Nope! That’s you adding a constraint that does not exist in the original problem. > Was Mr. Hall cheating? Not

> Also, Monty always opens a door with a goat. Nope! That’s you adding a constraint that does not exist in the original problem. > 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. From [1]. Constraints matter. Don’t play fast and loose with them. [1] https://www.nytimes.com/1991/07/21/us/behind-monty-hall-s-doors-puzzle-debate-and-answer.html https://www.nytimes.com/1991/07/21/us/behind-monty-hall-s-do...

ryao 2y ago
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.
- CrazyStat 2y ago
  You have hallucinated a constraint that never existed and then accused me of removing it.
  - ryao 2y ago
    The problem text itself specifies that the host opens an unselected door that has a goat behind it.
    - CrazyStat 2y ago
      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.
      - ryao 2y ago
        The problem text asks what is the better choice in this particular instance. It does not care about hypothetical other instances.
        CrazyStat 2y ago
        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.
        ryao 2y ago
        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.
    - penteract 2y ago
      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.
      - deleted 2y ago
        [deleted]
      - ryao 2y ago
        Here is what Marilyn vos Savant had to say: > 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. https://web.archive.org/web/20130121183432/http://marilynvossavant.com/game-show-problem/ https://web.archive.org/web/20130121183432/http://marilynvos... What you are discussing is a different problem.
        penteract 2y ago
        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).
        the_af 2y ago
        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.

ryao 2y ago

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.

CrazyStat 2y ago
You have hallucinated a constraint that never existed and then accused me of removing it.
- ryao 2y ago
  The problem text itself specifies that the host opens an unselected door that has a goat behind it.
  - CrazyStat 2y ago
    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.
    - ryao 2y ago
      The problem text asks what is the better choice in this particular instance. It does not care about hypothetical other instances.
      - CrazyStat 2y ago
        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.
        ryao 2y ago
        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.
  - penteract 2y ago
    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.
    - deleted 2y ago
      [deleted]
    - ryao 2y ago
      Here is what Marilyn vos Savant had to say: > 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. https://web.archive.org/web/20130121183432/http://marilynvossavant.com/game-show-problem/ https://web.archive.org/web/20130121183432/http://marilynvos... What you are discussing is a different problem.
      - penteract 2y ago
        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).
        the_af 2y ago
        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.