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Monty chooses a door? I thought he selected it at random. That was always the confusing unstated assumption for me, that makes me think the problem is only co

Monty chooses a door? I thought he selected it at random. That was always the confusing unstated assumption for me, that makes me think the problem is only confusing because of how the wording usually de-emphasizes that distinction. If he chooses a door, you can benefit from the secret information he reveals sometimes by doing so. If the door was chosen at random, he is not adding any information, so you can't act on it?

maxerickson 11y ago
He never reveals a car when he opens the door.
- malisper 11y ago
  That actually doesn't matter. What does matter is if he knows he is going to reveal a door that is empty. If he does know you should switch. If he reveals a door arbitrarily, and it just so happens to be empty, it doesn't matter if you switch. Edit: Since I have been downvoted, here is a citation: http://probability.ca/jeff/writing/montyfall.pdf http://probability.ca/jeff/writing/montyfall.pdf (see Monty Fall problem)
  - maxerickson 11y ago
    I agree that my phrasing is not superiorally precise, but the meaning is clear enough, the choice of the door isn't arbitrary.
  - kgwgk 11y ago
    To be fully pedantic, what matters is not "if he knows he is going to reveal a door that is empty". Maybe he has no idea and he just reads his lines off a teleprompter. What matters is if you know that he is going to reveal a door that is empty, because then you can reason according to that information.
  - baddox 11y ago
    The notion that he always randomly opens a door without knowing what's behind it but the door is always a losing door is fairly bizarre and doesn't make much sense when you try to analyze probability in the context of repeated experiments. Something must make him always choose a losing door. Either he knows what's behind the doors, or there's some external force ("fate," "God," or whatever) that causes the door to always be a losing door. Regardless, the problem as stated has Monty Hall open a losing door. Thus it's clear that you should switch after Monty Hall does so, and a very simple computer program can show that you tend to win by switching. Another way to phrase the problem is that you choose one door, then Monty Hall (without revealing anything) gives you the option of taking the door you chose, or taking both the other doors. That's an equivalent problem (assuming the goat has zero utility), and it makes it very obvious that you should switch.
    - dllthomas 11y ago
      "Something must make him always choose a losing door." The problem is not always presented in a way that makes it clear that the game always progresses this way, and it's not just a description of the particular circumstance you find yourself in one particular play. Consider: "You have $X, your opponents have $Y and $Z. You select Potent Potables for $400 and it's a daily double. How much should you wager?" I think you'll agree that is going to be read near-universally as a statement of a particular situation that could arise in Jeopardy, not a statement of how Jeopardy games always go.
      - baddox 11y ago
        > The problem is not always presented in a way that makes it clear that the game always progresses this way, and it's not just a description of the particular circumstance you find yourself in one particular play. I'm not sure how that matters. The question is whether, in this scenario, you choose to switch after Monty Hall reveals the door. The terms of the thought experiment dictate that Monty Hall will reveal a losing door. Regardless of what force is actually causing the revealed door to be a losing door, you should switch, because you're essentially being given the option to take your original one door, or to take both of the other doors (at least one of which is a losing door).
aborodin 11y ago
In the original statement of the problem, Monty knows what is behind each door. If he selected the door at random, the point of the problem is moot.
- proksoup 11y ago
  It's not just that he knows, but that he uses that knowledge to pick a door instead of choosing a door at random. But yeah, that the problem is moot if it were random is exactly my point. I was deceived by the wording and it's frustrating to hear "answers" to this problem that ignore that deception because I got the wrong answer and so I feel "dumb" for being decieved so I am trying to defend how i'm not "dumb" for not seeing the probability, just dumb for not thinking through the fact that it would be unlikely for Monty to reveal a car at that point. I guess a lot of people go "oh, he won't reveal a car". I'm missing that intuition.
tel 11y ago
That's the whole reason why the problem is tough. It's carefully worded to be ambiguous about that point so that most people assume it was random (most of the erroneous math people pull out is valid were it random) but the problem actually makes little sense if Monty picked randomly since it completely ignores the potential possibility that he picked the car door. The problem isn't probabilistically difficult and most people's intuitions would be on point if it weren't set up to deceive.
- JoeAltmaier 11y ago
  Ah. In my generation, who actually watched the program, there is no ambiguity. No intent to deceive was detected by me. Its not supposed to be a trick question. Its supposed to illustrate how far off our intuition is about statistical behavior. Monty himself mentions that he would occasionally try hard to argue contestants into swapping. They would always refuse. He would even explain that 'his door' had the better chance - nobody would listen. But I suppose that's confounded by the suspicion that he was trying to trick them.
  - proksoup 11y ago
    I agree it would help to see the problem in action. Again, the part I missed was "it would be silly for him to reveal a car". I get that now, but that was not something that I intuited, and so the omission of that information I will continue to argue is at least a little bit deceptive. Even if only to those like myself who have not seen the program.
  - deleted 11y ago
    [deleted]
  - tel 11y ago
    This may be very true and I think that if you're more familiar with the program and recognize that the host must know the answer then it becomes more a question of statistical intuition. I'm confident that there are lots of people who still fail to make the connection (though we're getting to the point where I think people are failing less due to a lack of statistical intuition and more due to a symbolic/physical model mismatch issue), but I think this problem wouldn't be as renowned as it is if it weren't for all of the even expert statisticians who are getting fooled. In their case it's definitely a matter of them misinterpreting the situation and the way that it's worded, for someone unfamiliar with the show, is at least a little "tricky".
dekhn 11y ago
This was the confusing unstated assumption for me, which is why I got so angry when Marilyn Vos Savant introduced the problem to me and many others here: https://en.wikipedia.org/wiki/Monty_Hall_problem#Vos_Savant_and_the_media_furor https://en.wikipedia.org/wiki/Monty_Hall_problem#Vos_Savant_... At the point when I read it, I didn't realize Monty was always choosing the door with pre-knowledge of which one the car wasn't behind. It wasn't until somebody wrote in with a computer program example that showed the benefits of switching, in which I inspected the source code, that I understand what Monty was doing. personally, I think this has to do with ambiguous way word problems are stated, and how people model the word problem mentally.
- adevine 11y ago
  Agreed, but the Monty Hall problem wouldn't make sense (at least from the game show perspective) if Monty's selection were random. If his selection were random, and he picked the car, it would just be "OK, I randomly decided to pick a door, and look, it's the car. Congratulations, game over."
  - baddox 11y ago
    Yeah, there really isn't any way to implement that version of the Monty Hall problem in the real world. You could say that Monty Hall randomly picks a door, and if he picks the winning door, that round of the game is aborted and you (the contestant) are given a drug that makes you forget that round ever happened. The rounds continue until Monty Hall happens to reveal a losing door, and that's the only one you will ever remember. This is so detached from normal events in life that the paradoxical nature is much less impactful, so it's nowhere near as enticing of a thought experiment. It is, however, similar (equivalent, methinks) to the "God's Coin Toss" problem, which is also popular and which also has gotten some attention on Hacker News: http://www.scottaaronson.com/democritus/lec17.html http://www.scottaaronson.com/democritus/lec17.html
    - dllthomas 11y ago
      Eh, the math works out the same whatever is done in the case of him revealing a car. Reasonable options seem to include: 1) contestant wins (notion being the contestant retains the option to switch to any door, and now knows where the car is) 2) contestant loses (notion being Monty picked right and "won" in place of the contestant) 3) Round is aborted, things are repositioned, and the round is replayed (doesn't require any drugging, aborted rounds may or may not be aired but player learned nothing relevant to future rounds).
jessaustin 11y ago
I know what you mean! I sort-of understand why people don't make this clear when they're stating the problem, but I really don't understand the vast majority of people who don't explain it clearly when they're stating the answer to the problem. They'll go on for paragraphs, when they could have just said, "Monty only chooses doors without cars, so by choosing a door he gave you more information."

maxerickson 11y ago

He never reveals a car when he opens the door.

malisper 11y ago
That actually doesn't matter. What does matter is if he knows he is going to reveal a door that is empty. If he does know you should switch. If he reveals a door arbitrarily, and it just so happens to be empty, it doesn't matter if you switch. Edit: Since I have been downvoted, here is a citation: http://probability.ca/jeff/writing/montyfall.pdf http://probability.ca/jeff/writing/montyfall.pdf (see Monty Fall problem)
- maxerickson 11y ago
  I agree that my phrasing is not superiorally precise, but the meaning is clear enough, the choice of the door isn't arbitrary.
- kgwgk 11y ago
  To be fully pedantic, what matters is not "if he knows he is going to reveal a door that is empty". Maybe he has no idea and he just reads his lines off a teleprompter. What matters is if you know that he is going to reveal a door that is empty, because then you can reason according to that information.
- baddox 11y ago
  The notion that he always randomly opens a door without knowing what's behind it but the door is always a losing door is fairly bizarre and doesn't make much sense when you try to analyze probability in the context of repeated experiments. Something must make him always choose a losing door. Either he knows what's behind the doors, or there's some external force ("fate," "God," or whatever) that causes the door to always be a losing door. Regardless, the problem as stated has Monty Hall open a losing door. Thus it's clear that you should switch after Monty Hall does so, and a very simple computer program can show that you tend to win by switching. Another way to phrase the problem is that you choose one door, then Monty Hall (without revealing anything) gives you the option of taking the door you chose, or taking both the other doors. That's an equivalent problem (assuming the goat has zero utility), and it makes it very obvious that you should switch.
  - dllthomas 11y ago
    "Something must make him always choose a losing door." The problem is not always presented in a way that makes it clear that the game always progresses this way, and it's not just a description of the particular circumstance you find yourself in one particular play. Consider: "You have $X, your opponents have $Y and $Z. You select Potent Potables for $400 and it's a daily double. How much should you wager?" I think you'll agree that is going to be read near-universally as a statement of a particular situation that could arise in Jeopardy, not a statement of how Jeopardy games always go.
    - baddox 11y ago
      > The problem is not always presented in a way that makes it clear that the game always progresses this way, and it's not just a description of the particular circumstance you find yourself in one particular play. I'm not sure how that matters. The question is whether, in this scenario, you choose to switch after Monty Hall reveals the door. The terms of the thought experiment dictate that Monty Hall will reveal a losing door. Regardless of what force is actually causing the revealed door to be a losing door, you should switch, because you're essentially being given the option to take your original one door, or to take both of the other doors (at least one of which is a losing door).

aborodin 11y ago

In the original statement of the problem, Monty knows what is behind each door. If he selected the door at random, the point of the problem is moot.

proksoup 11y ago
It's not just that he knows, but that he uses that knowledge to pick a door instead of choosing a door at random. But yeah, that the problem is moot if it were random is exactly my point. I was deceived by the wording and it's frustrating to hear "answers" to this problem that ignore that deception because I got the wrong answer and so I feel "dumb" for being decieved so I am trying to defend how i'm not "dumb" for not seeing the probability, just dumb for not thinking through the fact that it would be unlikely for Monty to reveal a car at that point. I guess a lot of people go "oh, he won't reveal a car". I'm missing that intuition.

tel 11y ago

That's the whole reason why the problem is tough. It's carefully worded to be ambiguous about that point so that most people assume it was random (most of the erroneous math people pull out is valid were it random) but the problem actually makes little sense if Monty picked randomly since it completely ignores the potential possibility that he picked the car door. The problem isn't probabilistically difficult and most people's intuitions would be on point if it weren't set up to deceive.

JoeAltmaier 11y ago
Ah. In my generation, who actually watched the program, there is no ambiguity. No intent to deceive was detected by me. Its not supposed to be a trick question. Its supposed to illustrate how far off our intuition is about statistical behavior. Monty himself mentions that he would occasionally try hard to argue contestants into swapping. They would always refuse. He would even explain that 'his door' had the better chance - nobody would listen. But I suppose that's confounded by the suspicion that he was trying to trick them.
- proksoup 11y ago
  I agree it would help to see the problem in action. Again, the part I missed was "it would be silly for him to reveal a car". I get that now, but that was not something that I intuited, and so the omission of that information I will continue to argue is at least a little bit deceptive. Even if only to those like myself who have not seen the program.
- deleted 11y ago
  [deleted]
- tel 11y ago
  This may be very true and I think that if you're more familiar with the program and recognize that the host must know the answer then it becomes more a question of statistical intuition. I'm confident that there are lots of people who still fail to make the connection (though we're getting to the point where I think people are failing less due to a lack of statistical intuition and more due to a symbolic/physical model mismatch issue), but I think this problem wouldn't be as renowned as it is if it weren't for all of the even expert statisticians who are getting fooled. In their case it's definitely a matter of them misinterpreting the situation and the way that it's worded, for someone unfamiliar with the show, is at least a little "tricky".

dekhn 11y ago

This was the confusing unstated assumption for me, which is why I got so angry when Marilyn Vos Savant introduced the problem to me and many others here: https://en.wikipedia.org/wiki/Monty_Hall_problem#Vos_Savant_and_the_media_furor https://en.wikipedia.org/wiki/Monty_Hall_problem#Vos_Savant_... At the point when I read it, I didn't realize Monty was always choosing the door with pre-knowledge of which one the car wasn't behind. It wasn't until somebody wrote in with a computer program example that showed the benefits of switching, in which I inspected the source code, that I understand what Monty was doing. personally, I think this has to do with ambiguous way word problems are stated, and how people model the word problem mentally.

adevine 11y ago
Agreed, but the Monty Hall problem wouldn't make sense (at least from the game show perspective) if Monty's selection were random. If his selection were random, and he picked the car, it would just be "OK, I randomly decided to pick a door, and look, it's the car. Congratulations, game over."
- baddox 11y ago
  Yeah, there really isn't any way to implement that version of the Monty Hall problem in the real world. You could say that Monty Hall randomly picks a door, and if he picks the winning door, that round of the game is aborted and you (the contestant) are given a drug that makes you forget that round ever happened. The rounds continue until Monty Hall happens to reveal a losing door, and that's the only one you will ever remember. This is so detached from normal events in life that the paradoxical nature is much less impactful, so it's nowhere near as enticing of a thought experiment. It is, however, similar (equivalent, methinks) to the "God's Coin Toss" problem, which is also popular and which also has gotten some attention on Hacker News: http://www.scottaaronson.com/democritus/lec17.html http://www.scottaaronson.com/democritus/lec17.html
  - dllthomas 11y ago
    Eh, the math works out the same whatever is done in the case of him revealing a car. Reasonable options seem to include: 1) contestant wins (notion being the contestant retains the option to switch to any door, and now knows where the car is) 2) contestant loses (notion being Monty picked right and "won" in place of the contestant) 3) Round is aborted, things are repositioned, and the round is replayed (doesn't require any drugging, aborted rounds may or may not be aired but player learned nothing relevant to future rounds).

jessaustin 11y ago

I know what you mean! I sort-of understand why people don't make this clear when they're stating the problem, but I really don't understand the vast majority of people who don't explain it clearly when they're stating the answer to the problem. They'll go on for paragraphs, when they could have just said, "Monty only chooses doors without cars, so by choosing a door he gave you more information."