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The main thing that seems to cause the paradox is a lack of specificity around the door opening mechanism. If you make it clear that Monty only opens a door tha

The main thing that seems to cause the paradox is a lack of specificity around the door opening mechanism. If you make it clear that Monty only opens a door that will definitely be empty, I find that most commentators will agree that Marilyn is correct, you should switch.

niel 7y ago
The original question that vos Savant responded to contains the phrase "...and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat..." - I don't know how they could be any more specific. If the host opens the door with the car, the game would be over anyway. It is clearly implied that the host always opens a door with a goat.
- tromp 7y ago
  > I don't know how they could be any more specific. "the host opens another door that he knows has a goat behind it, say No. 3," Using the knowledge of where the car is to avoid showing it, is more specific than just having the knowledge, and possibly still opening at random.
  - kgwgk 7y ago
    It could me more specific: “the host opens another door because he knows there is a goat behind it”.
  - niel 7y ago
    Whether the host opens a door with a goat or a car determines whether the game can still be played at all. How that door is selected does not factor into whether the contestant should switch or not in any way. I think the source of the paradox lies somewhere in the biases (the Endowment effect or Status Quo bias) as discussed in the Wikipedia article about the problem - not with how the question was stated and especially not with the part about how the host selects a door.
    - CJefferson 7y ago
      Ah, you are wrong (and this is often confusing to people). If you model this game where the host chooses randomly, and might open the car door (when that hapoens the player instantly loses) , then there is no point switching. I think the fact this does matter is the source of many people's confusion.
      - niel 7y ago
        > when that happens the player instantly loses I don't think I've ever seen this rule in any variants. Since we're making up rules: Could the player not still have the option to switch, and thus win the game? I'm not convinced that I am wrong after your explanation. Either way, my point is that focusing on the problem statement as the "main thing that seems to cause the paradox" allows the reader to stop there and disregard a far more interesting discussion about biases.
        OJFord 7y ago
        How is that rule made up? There's one car, if that door's been opened it's not left for the player to open. Obviously you can't win by switching to the location you know it's in because you've seen it, that's not a game.
      - Sandman 7y ago
        then there is no point switching Why not? Your chances of choosing the correct door were 1 in 3 from the start. That doesn't change with the fact that the host opens the doors at random.
        abiogenesis 7y ago
        That's what the GP said. In reality, the host doesn't open the door at random. They know which door hides the car and avoid opening it. That's why switching the doors doubles your chances of finding it (because the host has already eliminated one of the goat doors).
        7y ago
  - alvarelle 7y ago
    If the host knows where the car is, why would it ask the player to change door if he knows the player already choose a goat? The problem should specify that those are the rules of the game that the host always open a door. Otherwise, as a host, I would only offer the players to change if they picked the car. (And if they heard about this paradox before, they might even change their choice)
    - Sandman 7y ago
      If you only ever offered the players to switch if they picked the car, they'd soon realize this and never switch when they're offered this chance. Also, players that picked a goat at first would have no chance of winning the car.
      - alvarelle 7y ago
        That's assuming players knows about the other players, which is not specified in the problem description. It is not even specified if the same game is played several time with the same rules. Given the description from the article: > 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? I'd say, no, it is not to my advantage because I'd think the host would only ask me to switch if I had taken the "right" choice and want to make me lose. Unless I knew that the host always ask if one wants to change, in which case the paradox indeed apply.
        1123581321 7y ago
        Game show regulations would not allow that behavior. The problem assumes a typical game show in western media so the reader should assume the player knows all the rules governing the host’s behavior, which you can see in the other discussions.
- CJefferson 7y ago
  I don't think it is clearly implied. There is an old UK quiz show called "strike it lucky" where the host would show future symbols which might be good or bad for the players, rather than explictly showing "good" things.
- em500 7y ago
  The lack of specificity is not that the host might open a door with a car, but whether the host had to open a door at all. This is not clear from Savants formulation, and indeed does not match the actual game show, where the host had discretion in every episode whether to open a door. If the host does not have to open a door for the candidate (which is quite likely, he's playing his own strategy of making the show exciting, not predictable), then the usual solution (always switch) is not correct. All this, and probably any variation and every solution of the problem that comes up in this thread (and probably every other Monty Hall forum thread in the post 3 decades) are discussed extensively in the Wikipedia entry (e.g. https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_behaviors https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...). Maybe every time a new Monty Hall thread comes a mod should just stick the Wikipedia entry at the top. I've never seen anything in any of the following discussion that hasn't been covered in Wikipedia already.
mikekchar 7y ago
I don't think it's true that most people will agree. The problem is that intuitively it's easy to think that the chance of guessing the car is 1:3. That chance doesn't change just because you later know that another door has a goat. So you think that your choice was originally 1:3, the door that never gets revealed is 1:3 which are both equal. So when the other door is revealed, both chances move to 1:2. It's wrong, of course, but it's very easy to think that way (of course, I never thought that... ever... really... ha ha...)
- e12e 7y ago
  I think what's confusing, is that if one doesn't switch, the first selection is still 1:3 - since you randomly picked 1 door out of 3 (the fact that one is revealed does not change that). The strange thing is that "choosing to stay" somehow doesn't behave like a new 50/50 choice, but choosing to change does. Intuition implies that given a random choice between a, b and c - chances are 1:3. And given a new choice a or b, chances are 50/50. But you only get to act on new information (make a new choice) if you change. I must admit I'm still confused on this, and wonder if it bears out in simulation (that always change is better than random stay/change when offered to choose again Ed: paultopia has a nice simulation and explanation below: https://paul-gowder.com/montyhall/ https://paul-gowder.com/montyhall/ Significantly - staying means the player remains at 1:3, but since we now know the chosen door, or the alternate, holds the car - if staying is 1:3 wins, changing must account for the remaining 2:3 - since there are only two options. And random choice is also better than staying (it's 50/50) - but worse than changing.
  - tsimionescu 7y ago
    > Intuition implies that given a random choice between a, b and c - chances are 1:3. And given a new choice a or b, chances are 50/50. But you only get to act on new information (make a new choice) if you change. This is not the correct interpretation. In fact, the choice of getting the car if you switch is not 50/50, up from 1/3. It's 2/3. The choice of winning if you don't switch is 1/3. The reason for this is that the second choice is not independent of the first choice, so you shouldn't model them as different choices. Essentially the problem could be reformulated as: you choose one door at random. Then the host asks you if you would rather keep that door, or choose to get what is behind both of the other doors. The door opening is just a red herring.
peteretep 7y ago
> If you make it clear that Monty only opens a door that will definitely be empty, I find that most commentators will agree that Marilyn is correct, you should switch. I can’t argue with what you’ve observed, but my own experience is not that there’s a misunderstanding about the question, but that people simply don’t think properly about the question. Nobody I’ve ever seen confused about the answer was confused by the question, and many numerically literate people I know have struggled with it.

niel 7y ago

The original question that vos Savant responded to contains the phrase "...and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat..." - I don't know how they could be any more specific. If the host opens the door with the car, the game would be over anyway. It is clearly implied that the host always opens a door with a goat.

tromp 7y ago
> I don't know how they could be any more specific. "the host opens another door that he knows has a goat behind it, say No. 3," Using the knowledge of where the car is to avoid showing it, is more specific than just having the knowledge, and possibly still opening at random.
- kgwgk 7y ago
  It could me more specific: “the host opens another door because he knows there is a goat behind it”.
- niel 7y ago
  Whether the host opens a door with a goat or a car determines whether the game can still be played at all. How that door is selected does not factor into whether the contestant should switch or not in any way. I think the source of the paradox lies somewhere in the biases (the Endowment effect or Status Quo bias) as discussed in the Wikipedia article about the problem - not with how the question was stated and especially not with the part about how the host selects a door.
  - CJefferson 7y ago
    Ah, you are wrong (and this is often confusing to people). If you model this game where the host chooses randomly, and might open the car door (when that hapoens the player instantly loses) , then there is no point switching. I think the fact this does matter is the source of many people's confusion.
    - niel 7y ago
      > when that happens the player instantly loses I don't think I've ever seen this rule in any variants. Since we're making up rules: Could the player not still have the option to switch, and thus win the game? I'm not convinced that I am wrong after your explanation. Either way, my point is that focusing on the problem statement as the "main thing that seems to cause the paradox" allows the reader to stop there and disregard a far more interesting discussion about biases.
      - OJFord 7y ago
        How is that rule made up? There's one car, if that door's been opened it's not left for the player to open. Obviously you can't win by switching to the location you know it's in because you've seen it, that's not a game.
    - Sandman 7y ago
      then there is no point switching Why not? Your chances of choosing the correct door were 1 in 3 from the start. That doesn't change with the fact that the host opens the doors at random.
      - abiogenesis 7y ago
        That's what the GP said. In reality, the host doesn't open the door at random. They know which door hides the car and avoid opening it. That's why switching the doors doubles your chances of finding it (because the host has already eliminated one of the goat doors).
      - 7y ago
- alvarelle 7y ago
  If the host knows where the car is, why would it ask the player to change door if he knows the player already choose a goat? The problem should specify that those are the rules of the game that the host always open a door. Otherwise, as a host, I would only offer the players to change if they picked the car. (And if they heard about this paradox before, they might even change their choice)
  - Sandman 7y ago
    If you only ever offered the players to switch if they picked the car, they'd soon realize this and never switch when they're offered this chance. Also, players that picked a goat at first would have no chance of winning the car.
    - alvarelle 7y ago
      That's assuming players knows about the other players, which is not specified in the problem description. It is not even specified if the same game is played several time with the same rules. Given the description from the article: > 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? I'd say, no, it is not to my advantage because I'd think the host would only ask me to switch if I had taken the "right" choice and want to make me lose. Unless I knew that the host always ask if one wants to change, in which case the paradox indeed apply.
      - 1123581321 7y ago
        Game show regulations would not allow that behavior. The problem assumes a typical game show in western media so the reader should assume the player knows all the rules governing the host’s behavior, which you can see in the other discussions.
CJefferson 7y ago
I don't think it is clearly implied. There is an old UK quiz show called "strike it lucky" where the host would show future symbols which might be good or bad for the players, rather than explictly showing "good" things.
em500 7y ago
The lack of specificity is not that the host might open a door with a car, but whether the host had to open a door at all. This is not clear from Savants formulation, and indeed does not match the actual game show, where the host had discretion in every episode whether to open a door. If the host does not have to open a door for the candidate (which is quite likely, he's playing his own strategy of making the show exciting, not predictable), then the usual solution (always switch) is not correct. All this, and probably any variation and every solution of the problem that comes up in this thread (and probably every other Monty Hall forum thread in the post 3 decades) are discussed extensively in the Wikipedia entry (e.g. https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_behaviors https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...). Maybe every time a new Monty Hall thread comes a mod should just stick the Wikipedia entry at the top. I've never seen anything in any of the following discussion that hasn't been covered in Wikipedia already.

mikekchar 7y ago

I don't think it's true that most people will agree. The problem is that intuitively it's easy to think that the chance of guessing the car is 1:3. That chance doesn't change just because you later know that another door has a goat. So you think that your choice was originally 1:3, the door that never gets revealed is 1:3 which are both equal. So when the other door is revealed, both chances move to 1:2. It's wrong, of course, but it's very easy to think that way (of course, I never thought that... ever... really... ha ha...)

e12e 7y ago
I think what's confusing, is that if one doesn't switch, the first selection is still 1:3 - since you randomly picked 1 door out of 3 (the fact that one is revealed does not change that). The strange thing is that "choosing to stay" somehow doesn't behave like a new 50/50 choice, but choosing to change does. Intuition implies that given a random choice between a, b and c - chances are 1:3. And given a new choice a or b, chances are 50/50. But you only get to act on new information (make a new choice) if you change. I must admit I'm still confused on this, and wonder if it bears out in simulation (that always change is better than random stay/change when offered to choose again Ed: paultopia has a nice simulation and explanation below: https://paul-gowder.com/montyhall/ https://paul-gowder.com/montyhall/ Significantly - staying means the player remains at 1:3, but since we now know the chosen door, or the alternate, holds the car - if staying is 1:3 wins, changing must account for the remaining 2:3 - since there are only two options. And random choice is also better than staying (it's 50/50) - but worse than changing.
- tsimionescu 7y ago
  > Intuition implies that given a random choice between a, b and c - chances are 1:3. And given a new choice a or b, chances are 50/50. But you only get to act on new information (make a new choice) if you change. This is not the correct interpretation. In fact, the choice of getting the car if you switch is not 50/50, up from 1/3. It's 2/3. The choice of winning if you don't switch is 1/3. The reason for this is that the second choice is not independent of the first choice, so you shouldn't model them as different choices. Essentially the problem could be reformulated as: you choose one door at random. Then the host asks you if you would rather keep that door, or choose to get what is behind both of the other doors. The door opening is just a red herring.

peteretep 7y ago

> If you make it clear that Monty only opens a door that will definitely be empty, I find that most commentators will agree that Marilyn is correct, you should switch. I can’t argue with what you’ve observed, but my own experience is not that there’s a misunderstanding about the question, but that people simply don’t think properly about the question. Nobody I’ve ever seen confused about the answer was confused by the question, and many numerically literate people I know have struggled with it.