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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

by 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).
      - CJefferson 7y ago
        Your chance was 1 in 3 yes, but if the host can randomly open the car door, there is a chance of 1 in 3 of that happening, leaving only 1 in 3 chwnce of switching being legal and useful.
      - deeg 7y ago
        Sorry for the late reply, but if you look at the original article on GitHub I explore that scenario in the code. The answer is that if Monty randomly opens a door then the contestant automatically loses 1/3 of the time before given a chance to switch.
- 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.

by niel 7y ago

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).
    - CJefferson 7y ago
      Your chance was 1 in 3 yes, but if the host can randomly open the car door, there is a chance of 1 in 3 of that happening, leaving only 1 in 3 chwnce of switching being legal and useful.
    - deeg 7y ago
      Sorry for the late reply, but if you look at the original article on GitHub I explore that scenario in the code. The answer is that if Monty randomly opens a door then the contestant automatically loses 1/3 of the time before given a chance to switch.
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.