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> many formulations one finds (including the one in her column) As far as I’ve been able to find the text below would be the original question (“the host, who

by kgwgk 1y ago

> many formulations one finds (including the one in her column) As far as I’ve been able to find the text below would be the original question (“the host, who knows what’s behind the doors, opens”) and answer (“the host, who knows what’s behind the doors and will always avoid the one with the prize, opens”). Are you referring to that? https://web.archive.org/web/20130121183432/http://marilynvossavant.com/game-show-problem/ https://web.archive.org/web/20130121183432/http://marilynvos... 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 #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? [Craig F. Whitaker - Columbia, Maryland] Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

FabHK 1y ago
Yes, indeed, thanks for digging it up. You see that Marilyn's answer contains the clarified correct specification ("the host, who knows what’s behind the doors and will always avoid the one with the prize..."), but the question does not fully specify it. Thus, ignoring the clarification in the answer (which furthermore pertains to a modified problem), one could interpret the question differently: the host randomly opens a door [0], this specific day it happens to contain a goat, should you switch? And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory). [0] Note that specifying that the host knows what's behind what door doesn't help. They could still pick a door randomly.
- kgwgk 1y ago
  Did any of those readers of her column, including PhD and mathematicians, that declared her solution wrong do so because they objected to the clarification? (Does it make sense to declare the answer wrong ignoring the answer?)
  - FabHK 1y ago
    I think they didn't notice the significance of that clarification (which, again, was given in the context of a modified problem). Reading the exchanges again, though, I see that many respondents were not only mistaken, but also impolite or sexist in accusing Marylin of being wrong. If you specify the problem carefully, anyone with some training in probability should get the answer right. But clearly back then it was a head scratcher. I wonder whether there've been other problems like that, or we will encounter similar ones: elementary and easy to understand, yet many people get it confidently wrong, until the correct solution permeates through culture.
    - kgwgk 1y ago
      > [that clarification] was given in the context of a modified problem I don’t understand what do you mean by that. The clarification was part of the original answer in the context of the original question. The clarification was stated again and again and again. If that’s a “modified problem” so be it but that was the problem being discussed with her readers.
      - FabHK 1y ago
        The modification I'm referring to is the one Marylin suggests with a million doors, with Monty opening 999,998 of.
        kgwgk 1y ago
        Ok, so the modification was the "Here’s a good way to visualize what happened." explanation. I would say that the whole (short) paragraph is intended to clarify why the answer to the question is "Yes; you should switch." Maybe it was not a good way to visualize what happened after all. Experience shows that some readers were unable to understand the argument. However, none of them complained about the explanation being wrong because those were two completely different problems (only in the second one the host will always avoid the door with the prize, apparently).
    - SAI_Peregrinus 1y ago
- fenomas 1y ago
  I don't think this complaint stands at all. The original question is stated colloquially, not formally, and vos Savant's interpretation of it is the one that's consistent with the question's language and with how game shows work. Even if other interpretations are possible, hers is the one a normal reader would make, and the one the letter-writer apparently intends. > And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory). I don't think this stands either - the letters quoted don't say anything about the distinction you're making. Unless they're all fictional or selectively edited, a bunch of PhDs really did get the puzzle wrong.
- bmm6o 1y ago
  There's also the cultural context of Monty Hall being a real person who had a real game show, on which he really opened doors with goats behind them. Most readers of her column would have been familiar with the mechanics of the show. And the question doesn't really make sense if there's a chance that he opens the door with a prize, there's no more hidden information in that case.
michaelcampbell 1y ago
> Suppose there are a million doors,... This is a common way to explain it, but I always found it less intuitive; not sure why. My go to thought experiment along these lines is rather this: Adjust the game to where both you and your friend are playing at the same time. You _NEVER_ switch doors, and your friend _ALWAYS_ does, when given the choice by Monty. Since you never switch, no matter what Monty does you know you're going to win 1/3 of the time. Since you both know Monty is always going to show a "goat" door, ONE of you must win, so your friend MUST WIN ALL THE TIMES YOU DO NOT. Since you win 1/3 of the time, the wins "left" is 2/3 of the time.
- taid9iK- 1y ago
  Oh I like that one. I am having real trouble with the commonly used explanations.

by kgwgk 1y ago

FabHK 1y ago

Yes, indeed, thanks for digging it up. You see that Marilyn's answer contains the clarified correct specification ("the host, who knows what’s behind the doors and will always avoid the one with the prize..."), but the question does not fully specify it. Thus, ignoring the clarification in the answer (which furthermore pertains to a modified problem), one could interpret the question differently: the host randomly opens a door [0], this specific day it happens to contain a goat, should you switch? And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory). [0] Note that specifying that the host knows what's behind what door doesn't help. They could still pick a door randomly.

kgwgk 1y ago
Did any of those readers of her column, including PhD and mathematicians, that declared her solution wrong do so because they objected to the clarification? (Does it make sense to declare the answer wrong ignoring the answer?)
- FabHK 1y ago
  I think they didn't notice the significance of that clarification (which, again, was given in the context of a modified problem). Reading the exchanges again, though, I see that many respondents were not only mistaken, but also impolite or sexist in accusing Marylin of being wrong. If you specify the problem carefully, anyone with some training in probability should get the answer right. But clearly back then it was a head scratcher. I wonder whether there've been other problems like that, or we will encounter similar ones: elementary and easy to understand, yet many people get it confidently wrong, until the correct solution permeates through culture.
  - kgwgk 1y ago
    > [that clarification] was given in the context of a modified problem I don’t understand what do you mean by that. The clarification was part of the original answer in the context of the original question. The clarification was stated again and again and again. If that’s a “modified problem” so be it but that was the problem being discussed with her readers.
    - FabHK 1y ago
      The modification I'm referring to is the one Marylin suggests with a million doors, with Monty opening 999,998 of.
      - kgwgk 1y ago
        Ok, so the modification was the "Here’s a good way to visualize what happened." explanation. I would say that the whole (short) paragraph is intended to clarify why the answer to the question is "Yes; you should switch." Maybe it was not a good way to visualize what happened after all. Experience shows that some readers were unable to understand the argument. However, none of them complained about the explanation being wrong because those were two completely different problems (only in the second one the host will always avoid the door with the prize, apparently).
  - SAI_Peregrinus 1y ago
fenomas 1y ago
I don't think this complaint stands at all. The original question is stated colloquially, not formally, and vos Savant's interpretation of it is the one that's consistent with the question's language and with how game shows work. Even if other interpretations are possible, hers is the one a normal reader would make, and the one the letter-writer apparently intends. > And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory). I don't think this stands either - the letters quoted don't say anything about the distinction you're making. Unless they're all fictional or selectively edited, a bunch of PhDs really did get the puzzle wrong.
bmm6o 1y ago
There's also the cultural context of Monty Hall being a real person who had a real game show, on which he really opened doors with goats behind them. Most readers of her column would have been familiar with the mechanics of the show. And the question doesn't really make sense if there's a chance that he opens the door with a prize, there's no more hidden information in that case.

michaelcampbell 1y ago

> Suppose there are a million doors,... This is a common way to explain it, but I always found it less intuitive; not sure why. My go to thought experiment along these lines is rather this: Adjust the game to where both you and your friend are playing at the same time. You _NEVER_ switch doors, and your friend _ALWAYS_ does, when given the choice by Monty. Since you never switch, no matter what Monty does you know you're going to win 1/3 of the time. Since you both know Monty is always going to show a "goat" door, ONE of you must win, so your friend MUST WIN ALL THE TIMES YOU DO NOT. Since you win 1/3 of the time, the wins "left" is 2/3 of the time.

taid9iK- 1y ago
Oh I like that one. I am having real trouble with the commonly used explanations.