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I can assure you Diaconis didn’t get it wrong. > "The problem is not well-formed," Mr. Gardner said, "unless it makes clear that the host must always open an e

I can assure you Diaconis didn’t get it wrong. > "The problem is not well-formed," Mr. Gardner said, "unless it makes clear that the host must always open an empty door and offer the switch. Otherwise, if the host is malevolent, he may open another door only when it's to his advantage to let the player switch, and the probability of being right by switching could be as low as zero." Mr. Gardner said the ambiguity could be eliminated if the host promised ahead of time to open another door and then offer a switch. The hosts’s intentions absolutely do matter, because the problem (as originally stated) doesn’t specify that the host always opens a door and offers a switch. Maybe he only offers a trade when you initially picked the good door.

thadt 2y ago
> Maybe he only offers a trade when you initially picked the good door. That would be a rather convenient signal to the player.
- CrazyStat 2y ago
  Indeed. But the hosts machinations can be arbitrarily more complex; maybe he offers the switch to contestants he finds attractive only when they’ve picked a goat, and contestants he finds unattractive when they’ve picked the car.
  - the_af 2y ago
    You're introducing bizarre ad hoc hypotheses. This works as a logic puzzle. Assuming the host offers different doors depending on contestant attractiveness makes absolutely no sense. It's a bizarre assumption. Maybe the goats can wander from door to door, or maybe there is no car, or maybe behind all of the doors there are tigers. Which would be absurd and unrelated to this puzzle.
    - roenxi 2y ago
      The hosts' strategy is the core of the puzzle. The question is what information he has just conveyed to the player by opening the door and that depends entirely on his mental state. If he was always going to open a door then the player should switch. If he is opening a door only if the player has picked the car then they should not switch. If he has bizarre ad hoc motivations then the correct decision depends on bizarre ad hoc considerations. And, as CrazyStat has correctly pointed out, as stated in the linked article the hosts' strategy is an unknown. It could be bizarre. Although I'd still rather say vos Savant was correct in her reasoning; since the answer is interesting it seems fairer to blame the person posing the question for getting a detail wrong.
      - ryao 2y ago
        The source material for the article says otherwise: > 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...
        roenxi 2y ago
        You need to know why the host did that though. The host might have an adversarial strategy where they only open the door when using vos Savant's logic would make the player lose. Her logic was really interesting, it is easy to see why she gets to write a column. But at the end of the day the problem is technically ill-formed. > There’s no way he can always open a losing door by chance! Yes there is, he might have picked a remaining door at random with the plan of saying "You lose!" if he finds the car. An actor with perfect knowledge can still have a probabilistic strategy. That doesn't really change the decision to switch, but it does have a material impact on the analysis logic. She's correct that is a necessary assumption to get the most interesting form of the problem. But that isn't what the questioner asked.
        2y ago
  - ryao 2y ago
    This is like being asked how to solve a (legal) Rubik cube configuration and then considering how close you can come to a solution if given an illegal configuration. It is not relevant since it is not what was presented. You can always make things more complex by considering variations that are not relevant to the original problem.
  - robertlagrant 2y ago
    This was a real life gameshow with known rules. The host always opened a door, and always a door with a goat.
    - CrazyStat 2y ago
      This was a real life game show where the host did not always open a door and offer a switch. In fact most of the time he did not offer a switch. See [1] (Ctrl-F cheating for the relevant paragraph). [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
The problem as stated does not give the host such an option. If it did, the host opening a door would imply that the player picked the right answer, and it would only happen 1/3 of the time. Some of the comments aged fairly well, although not in the way that their authors intended: > There is enough mathematical illiteracy in this country > If all those Ph.D.’s were wrong, the country would be in some very serious trouble. In the 1800s, Carl Friedrich Gauss lamented about the decline in mathematical ability in academia. Despite academia since having advanced mathematics farther, mathematical ability in academia still has evidence of decline. Professors tend to be good at extremely specialized things, yet they get the simple things wrong. I once had a Calculus professor who failed to perform basic arithmetic correctly, during his calculus class. All of the algebra was right, but his constants were wrong. This happened on multiple occasions.
- CrazyStat 2y ago
  The problem as stated, in the article you pulled your quote from, puts no limits on how the host decides whether or not to offer a switch: > Imagine that you’re on a television game show and the host presents you with three closed doors. Behind one of them, sits a sparkling, brand-new Lincoln Continental; behind the other two, are smelly old goats. The host implores you to pick a door, and you select door #1. Then, the host, who is well-aware of what’s going on behind the scenes, opens door #3, revealing one of the goats. > “Now,” he says, turning toward you, “do you want to keep door #1, or do you want to switch to door #2?” All you know is that in this particular instance the host has opened a door and offered a switch. You cannot conclude that the host always opens a door and offers a switch. The problem as stated allows the host to offer switches only when the contestant picked the door with the prize, or only when the moon is gibbous, or only when the tide is going out. Diaconis and Gardner are completely correct to point out that the problem as stated is under specified and that the intent of the host matters.
  - ryao 2y ago
    The problem as stated has the host open an incorrect door and offer the player a chance to change his choice. Inferring that another possible variation might exist does not change the fact that we are discussing the variation that was presented. Both you and Diaconis are wrong.
the_af 2y ago
> [...] the problem (as originally stated) doesn’t specify that the host always opens a door and offers a switch. Maybe he only offers a trade when you initially picked the good door. That would make no sense. Also, Monty always opens a door with a goat. The problem is well-formed, but most people misrepresent it in order to object to it.
- CrazyStat 2y ago
  > 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.

thadt 2y ago

> Maybe he only offers a trade when you initially picked the good door. That would be a rather convenient signal to the player.

CrazyStat 2y ago
Indeed. But the hosts machinations can be arbitrarily more complex; maybe he offers the switch to contestants he finds attractive only when they’ve picked a goat, and contestants he finds unattractive when they’ve picked the car.
- the_af 2y ago
  You're introducing bizarre ad hoc hypotheses. This works as a logic puzzle. Assuming the host offers different doors depending on contestant attractiveness makes absolutely no sense. It's a bizarre assumption. Maybe the goats can wander from door to door, or maybe there is no car, or maybe behind all of the doors there are tigers. Which would be absurd and unrelated to this puzzle.
  - roenxi 2y ago
    The hosts' strategy is the core of the puzzle. The question is what information he has just conveyed to the player by opening the door and that depends entirely on his mental state. If he was always going to open a door then the player should switch. If he is opening a door only if the player has picked the car then they should not switch. If he has bizarre ad hoc motivations then the correct decision depends on bizarre ad hoc considerations. And, as CrazyStat has correctly pointed out, as stated in the linked article the hosts' strategy is an unknown. It could be bizarre. Although I'd still rather say vos Savant was correct in her reasoning; since the answer is interesting it seems fairer to blame the person posing the question for getting a detail wrong.
    - ryao 2y ago
      The source material for the article says otherwise: > 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...
      - roenxi 2y ago
        You need to know why the host did that though. The host might have an adversarial strategy where they only open the door when using vos Savant's logic would make the player lose. Her logic was really interesting, it is easy to see why she gets to write a column. But at the end of the day the problem is technically ill-formed. > There’s no way he can always open a losing door by chance! Yes there is, he might have picked a remaining door at random with the plan of saying "You lose!" if he finds the car. An actor with perfect knowledge can still have a probabilistic strategy. That doesn't really change the decision to switch, but it does have a material impact on the analysis logic. She's correct that is a necessary assumption to get the most interesting form of the problem. But that isn't what the questioner asked.
        2y ago
- ryao 2y ago
  This is like being asked how to solve a (legal) Rubik cube configuration and then considering how close you can come to a solution if given an illegal configuration. It is not relevant since it is not what was presented. You can always make things more complex by considering variations that are not relevant to the original problem.
- robertlagrant 2y ago
  This was a real life gameshow with known rules. The host always opened a door, and always a door with a goat.
  - CrazyStat 2y ago
    This was a real life game show where the host did not always open a door and offer a switch. In fact most of the time he did not offer a switch. See [1] (Ctrl-F cheating for the relevant paragraph). [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

The problem as stated does not give the host such an option. If it did, the host opening a door would imply that the player picked the right answer, and it would only happen 1/3 of the time. Some of the comments aged fairly well, although not in the way that their authors intended: > There is enough mathematical illiteracy in this country > If all those Ph.D.’s were wrong, the country would be in some very serious trouble. In the 1800s, Carl Friedrich Gauss lamented about the decline in mathematical ability in academia. Despite academia since having advanced mathematics farther, mathematical ability in academia still has evidence of decline. Professors tend to be good at extremely specialized things, yet they get the simple things wrong. I once had a Calculus professor who failed to perform basic arithmetic correctly, during his calculus class. All of the algebra was right, but his constants were wrong. This happened on multiple occasions.

CrazyStat 2y ago
The problem as stated, in the article you pulled your quote from, puts no limits on how the host decides whether or not to offer a switch: > Imagine that you’re on a television game show and the host presents you with three closed doors. Behind one of them, sits a sparkling, brand-new Lincoln Continental; behind the other two, are smelly old goats. The host implores you to pick a door, and you select door #1. Then, the host, who is well-aware of what’s going on behind the scenes, opens door #3, revealing one of the goats. > “Now,” he says, turning toward you, “do you want to keep door #1, or do you want to switch to door #2?” All you know is that in this particular instance the host has opened a door and offered a switch. You cannot conclude that the host always opens a door and offers a switch. The problem as stated allows the host to offer switches only when the contestant picked the door with the prize, or only when the moon is gibbous, or only when the tide is going out. Diaconis and Gardner are completely correct to point out that the problem as stated is under specified and that the intent of the host matters.
- ryao 2y ago
  The problem as stated has the host open an incorrect door and offer the player a chance to change his choice. Inferring that another possible variation might exist does not change the fact that we are discussing the variation that was presented. Both you and Diaconis are wrong.

the_af 2y ago

> [...] the problem (as originally stated) doesn’t specify that the host always opens a door and offers a switch. Maybe he only offers a trade when you initially picked the good door. That would make no sense. Also, Monty always opens a door with a goat. The problem is well-formed, but most people misrepresent it in order to object to it.

CrazyStat 2y ago
> 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.