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It doesn't matter at all if Monty knows which doors are winners and losers. If Monty doesn't know, then sometimes the game will be ruined because he will expos

It doesn't matter at all if Monty knows which doors are winners and losers. If Monty doesn't know, then sometimes the game will be ruined because he will expose the grand prize and then the game is moot. But if is simply lucky by showing the goat door vs he picked it with foreknowledge doesn't change the odds in any way.

lqet 6y ago
It does matter. If you are looking at a single instance where Monty got lucky and chose a goat door, then the probability that the door you have chosen has the car is 1/2, and switching doesn't change anything. This can easily be tested: just write a simulation that runs the experiment with Monty choosing a random door, and discard the instances where the game was "ruined" because he picked the car. In the remaining instances, both strategies will perform the same.
- Taek 6y ago
  It matters because in the case where Monty does know to avoid the door, those games end up as wins for the person who selected to switch, as opposed to broken games. You are taking an empty result and replacing it with a strict win.
  - SamBam 6y ago
    I think lqet's point is that if it's just recounted as being thrown into a situation -- you've picked a door, and Monty opens a door, and it's a goat -- you don't know if you're in a world where Monty always picks a goat or one where he just happened to be lucky. In the second scenario, it doesn't matter if you switch.
    - couchand 6y ago
      Well in that situation it's still beneficial to switch, unless the probability of being in the first scenario is exactly zero.
      - ubercow13 6y ago
        You’re forgetting about the possibility of being in a universe where Monty only opens a door if you chose the correct door originally (otherwise he does something else like ending the game). If you know nothing about the rule set of the game show, Monty opening the door tells you nothing and you have no reason to either switch or not switch.
      - SamBam 6y ago
        Indeed. It's like the argument that you may as well believe in God, because if he doesn't exist it doesn't hurt you. That said, though, if you're trying to convince someone of the correctness of the Monty Hall Problem, and they're stuck on that misconception, saying "well you may as well switch in case I'm right" isn't a great argument.
- basch 6y ago
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- jholman 6y ago
  Sure, here's the simulation. There are only six distinct cases, so we can "simulate" it exhaustively, and a Monte Carlo will approximate this. Let "A", "B", and "C" represent the prizes; A is valuable, B and C are goats. They're shuffled behind doors, so the player and Monty choose "1" or "2" or "3", and after the fact we'll map ABC to 123. As such, we can assume without loss of generality that the player always chooses "1" and Monty always chooses "2". What happens? 123 ABC - player was right at first, so they lose if they switch ACB - player was right at first, so they lose if they switch BAC - player was wrong, Monty chooses 2 which is the car, so the game is ruined BCA - player was wrong, Monty finds a goat, player wins if they switch. CAB - player was wrong, Monty chooses 2 which is the car, so the game is ruined CBA - player was wrong, Monty finds a goat, player wins if they switch. Huh. Of the 4 non-ruined games, half of them are improved by switching, half are worsened. I believed you were wrong, but now I believe you.
  - bonoboTP 6y ago
    Exactly and if Monty knows of the car location (as in the original formulation), he will basically save BAC and CAB from being ruined and both of these will become "player wins if they switch" outcomes. So instead of 2 out of 4, we now have 4 out of 6 cases where switching is better.
- graeme 6y ago
  Can’t that be expressed even more simply as: 1/3: you picked car, switch fails 1/3: you picked goat, monty ruins. Not included 1/3: you picked goat, monty open goat. Switch wins. Of the two non ruined scenarios, half win We can also demonstrate by using the 100 door example. Monty opens 98 of the 99 doors you didn’t choose. 98 times out of 99, monty ruins the game. So random choice dramatically changes things.
  - tasty_freeze 6y ago
    I think where I differ (also expressed on the parent comment) is that when Monty ruins the game, you are tossing out many cases where the "I'm sticking with my original choice" strategy failed, while I'm counting them as a failure in the strategy.
    - irishsultan 6y ago
      You are free to count them as a failure of the "I'm sticking with my original choice" strategy, but only if you also count them as a failure of the "I'm switching" strategy, because when Monty ruins the game there is no way to win the car by switching to the remaining closed door. If Monty chooses randomly both strategies fail 2/3 of the time.
      - graeme 6y ago
        Precisely. Either way switch or no switch have the same success rate if Monty ruins the game. You can count it as 1/2 or 2/3 but either way both choices are the same. So it’s a very different scenario from when Monty knows the door is a goat.
- tasty_freeze 6y ago
  Why are we discarding the games that are ruined? Whether Monty knows for sure or is just picking at random only affects whether games are ruined, not the probability that the door the contestant picked is a winner. If there are N doors and the player picks door 1, there is a 1/N chance he will be correct. If he sticks to that door, there is nothing that will ever change that 1/N odds that it was the winner. Downthread someone enumerated the cases, but again, they discard the 1/3 of the cases where Monty ruined it. What that does is throw out half of the cases where contestant picked the wrong door and had the wrong strategy. Whether Monty ruined the surprise by showing the winning door or Monty got lucky and exposed a goat, that original door always has a 1/3 probability of being the right one.
  - kgwgk 6y ago
    Because the problem asks what to do after a goat is shown.
  - ubercow13 6y ago
    >Whether Monty ruined the surprise by showing the winning door or Monty got lucky and exposed a goat, that original door always has a 1/3 probability of being the right one. By that logic, so does the remaining door. 1/3 vs 1/3.
  - bart_spoon 6y ago
    Because that is a fundamental part of the "Monty Hall" problem. You are correct in the case where Monty will randomly pick doors, and sometimes he will pick the winning door and ruin the game, but that is a fundamentally different problem than what is described as the "Monty Hall problem", and therefore irrelevant. Its like saying soccer players could score more if they picked up the ball and ran with it. While true, you are no longer describing the game of soccer.
dllthomas 6y ago
Once upon a time I was convinced of this, for reasons. I wrote a simulation to demonstrate how right I was. I was wrong and you are too.
- Tomminn 6y ago
  It's weirdly great that you had enough misplaced conviction to motivate you to write a simulation which ended up proving you wrong. It's actually an interesting, possibly general story, that kind of makes me think. Perhaps for certain personality types, misplaced conviction will put you on a higher velocity trajectory toward truth than honest confusion.
  - JauntyHatAngle 6y ago
    >Perhaps for certain personality types, misplaced conviction will put you on a higher velocity trajectory toward truth than honest confusion. The key part of this sentence that makes it true, is the "for certain personality types". I would wager the majority of personality types however, it would put you on a higher velocity trajectory to willful ignorance.
    - Tomminn 6y ago
      For sure. It's a small fraction of the population whose first instinct to prove themselves right is to do which also has the possibility of also proving themselves wrong.
  - chrisweekly 6y ago
    It's the story of science! Unlike lawyers or debaters, scientists are delighted when their hypotheses are clearly disproven; this is at the crux of the scientific method, and is essential for discovery of the truth. Assert something falsifiable, then experiment. The truth wins.
furyofantares 6y ago
In terms of the logic problem, I can't see any distinction between not counting moot games where he reveals a grand prize vs assuming he never reveals the grand prize. It's just two different ways to say that the only games under consideration are the ones where Monty reveals a goat. So I'm not getting the point you're making, unless you're saying that would be a more grokable way to state the problem?
- deleted 6y ago
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- qu4z-2 6y ago
  It makes a difference how you got there, assuming in the second one that he reliably/intentionally never reveals the grand prize. Imagine you have two identical looking bags of marbles, one contains two red marbles, and one contains one red and one black. You reach into a random one and draw a marble. It is red. If you picked the marble randomly, you now know you're more likely to be holding the bag that contained two red marbles than the mixed bag. The red marble you randomly drew provides bayesian evidence for which bag you're holding. Marble 1 2 Bag 2R R R BR B R Out of the three possible equally likely worlds (given that you know you didn't roll into BR-1), two of them have you holding bag 2R, aka a 2/3rds chance. If, on the other hand, you use a red-marble-picking robot, you get: Marble 1 2 Bag 2R R R BR R R Now you have four possible worlds and are equally likely to be holding either bag, ie 1/2. So, if our friend Monty opens a random door and you only look at simulations where it reveals a goat... There is a 1/3rd chance you initially get the car, but if you do he's guaranteed to reveal a goat. So 1/3rd of games, you have a car. There's a 2/3rd chance you initially get a goat, but then he has only a 50/50 chance of revealing the goat. So half of the 2/3rds of games, the game is discard as invalid, and the other half of 2/3rds (aka 1/3) you have a goat. As such, the potential outcomes are split 1:1:1 between car:goat:invalid. Removing the invalid cases, it's 1:1 car:goat, despite the only 1/3rd chance of you choosing the car initially. tl;dr: If he's randomly opening doors and you discard any where it's the car, those cases are entirely ones where you didn't initially choose the car, which introduces some bias. If he's opening doors deliberately, no cases are thrown out since it's always possible to reveal a goat, maintaining the original 1:2 probability ratio. EDIT: formatting.
Tomminn 6y ago
You only remove games as moot when Monty reveals a car, which can only happen when you haven't initially selected the car. Therefore, no games are removed as moot when you have initially selected a car, and 50% of games are removed as moot when you have initially selected a goat. So it ends in a coin toss whether you switch or not. The 2/3 to 1/3 split only happens when Monty removes a goat using privileged information.

lqet 6y ago

It does matter. If you are looking at a single instance where Monty got lucky and chose a goat door, then the probability that the door you have chosen has the car is 1/2, and switching doesn't change anything. This can easily be tested: just write a simulation that runs the experiment with Monty choosing a random door, and discard the instances where the game was "ruined" because he picked the car. In the remaining instances, both strategies will perform the same.

Taek 6y ago
It matters because in the case where Monty does know to avoid the door, those games end up as wins for the person who selected to switch, as opposed to broken games. You are taking an empty result and replacing it with a strict win.
- SamBam 6y ago
  I think lqet's point is that if it's just recounted as being thrown into a situation -- you've picked a door, and Monty opens a door, and it's a goat -- you don't know if you're in a world where Monty always picks a goat or one where he just happened to be lucky. In the second scenario, it doesn't matter if you switch.
  - couchand 6y ago
    Well in that situation it's still beneficial to switch, unless the probability of being in the first scenario is exactly zero.
    - ubercow13 6y ago
      You’re forgetting about the possibility of being in a universe where Monty only opens a door if you chose the correct door originally (otherwise he does something else like ending the game). If you know nothing about the rule set of the game show, Monty opening the door tells you nothing and you have no reason to either switch or not switch.
    - SamBam 6y ago
      Indeed. It's like the argument that you may as well believe in God, because if he doesn't exist it doesn't hurt you. That said, though, if you're trying to convince someone of the correctness of the Monty Hall Problem, and they're stuck on that misconception, saying "well you may as well switch in case I'm right" isn't a great argument.
basch 6y ago
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jholman 6y ago
Sure, here's the simulation. There are only six distinct cases, so we can "simulate" it exhaustively, and a Monte Carlo will approximate this. Let "A", "B", and "C" represent the prizes; A is valuable, B and C are goats. They're shuffled behind doors, so the player and Monty choose "1" or "2" or "3", and after the fact we'll map ABC to 123. As such, we can assume without loss of generality that the player always chooses "1" and Monty always chooses "2". What happens? 123 ABC - player was right at first, so they lose if they switch ACB - player was right at first, so they lose if they switch BAC - player was wrong, Monty chooses 2 which is the car, so the game is ruined BCA - player was wrong, Monty finds a goat, player wins if they switch. CAB - player was wrong, Monty chooses 2 which is the car, so the game is ruined CBA - player was wrong, Monty finds a goat, player wins if they switch. Huh. Of the 4 non-ruined games, half of them are improved by switching, half are worsened. I believed you were wrong, but now I believe you.
- bonoboTP 6y ago
  Exactly and if Monty knows of the car location (as in the original formulation), he will basically save BAC and CAB from being ruined and both of these will become "player wins if they switch" outcomes. So instead of 2 out of 4, we now have 4 out of 6 cases where switching is better.
graeme 6y ago
Can’t that be expressed even more simply as: 1/3: you picked car, switch fails 1/3: you picked goat, monty ruins. Not included 1/3: you picked goat, monty open goat. Switch wins. Of the two non ruined scenarios, half win We can also demonstrate by using the 100 door example. Monty opens 98 of the 99 doors you didn’t choose. 98 times out of 99, monty ruins the game. So random choice dramatically changes things.
- tasty_freeze 6y ago
  I think where I differ (also expressed on the parent comment) is that when Monty ruins the game, you are tossing out many cases where the "I'm sticking with my original choice" strategy failed, while I'm counting them as a failure in the strategy.
  - irishsultan 6y ago
    You are free to count them as a failure of the "I'm sticking with my original choice" strategy, but only if you also count them as a failure of the "I'm switching" strategy, because when Monty ruins the game there is no way to win the car by switching to the remaining closed door. If Monty chooses randomly both strategies fail 2/3 of the time.
    - graeme 6y ago
      Precisely. Either way switch or no switch have the same success rate if Monty ruins the game. You can count it as 1/2 or 2/3 but either way both choices are the same. So it’s a very different scenario from when Monty knows the door is a goat.
tasty_freeze 6y ago
Why are we discarding the games that are ruined? Whether Monty knows for sure or is just picking at random only affects whether games are ruined, not the probability that the door the contestant picked is a winner. If there are N doors and the player picks door 1, there is a 1/N chance he will be correct. If he sticks to that door, there is nothing that will ever change that 1/N odds that it was the winner. Downthread someone enumerated the cases, but again, they discard the 1/3 of the cases where Monty ruined it. What that does is throw out half of the cases where contestant picked the wrong door and had the wrong strategy. Whether Monty ruined the surprise by showing the winning door or Monty got lucky and exposed a goat, that original door always has a 1/3 probability of being the right one.
- kgwgk 6y ago
  Because the problem asks what to do after a goat is shown.
- ubercow13 6y ago
  >Whether Monty ruined the surprise by showing the winning door or Monty got lucky and exposed a goat, that original door always has a 1/3 probability of being the right one. By that logic, so does the remaining door. 1/3 vs 1/3.
- bart_spoon 6y ago
  Because that is a fundamental part of the "Monty Hall" problem. You are correct in the case where Monty will randomly pick doors, and sometimes he will pick the winning door and ruin the game, but that is a fundamentally different problem than what is described as the "Monty Hall problem", and therefore irrelevant. Its like saying soccer players could score more if they picked up the ball and ran with it. While true, you are no longer describing the game of soccer.

dllthomas 6y ago

Once upon a time I was convinced of this, for reasons. I wrote a simulation to demonstrate how right I was. I was wrong and you are too.

Tomminn 6y ago
It's weirdly great that you had enough misplaced conviction to motivate you to write a simulation which ended up proving you wrong. It's actually an interesting, possibly general story, that kind of makes me think. Perhaps for certain personality types, misplaced conviction will put you on a higher velocity trajectory toward truth than honest confusion.
- JauntyHatAngle 6y ago
  >Perhaps for certain personality types, misplaced conviction will put you on a higher velocity trajectory toward truth than honest confusion. The key part of this sentence that makes it true, is the "for certain personality types". I would wager the majority of personality types however, it would put you on a higher velocity trajectory to willful ignorance.
  - Tomminn 6y ago
    For sure. It's a small fraction of the population whose first instinct to prove themselves right is to do which also has the possibility of also proving themselves wrong.
- chrisweekly 6y ago
  It's the story of science! Unlike lawyers or debaters, scientists are delighted when their hypotheses are clearly disproven; this is at the crux of the scientific method, and is essential for discovery of the truth. Assert something falsifiable, then experiment. The truth wins.

furyofantares 6y ago

In terms of the logic problem, I can't see any distinction between not counting moot games where he reveals a grand prize vs assuming he never reveals the grand prize. It's just two different ways to say that the only games under consideration are the ones where Monty reveals a goat. So I'm not getting the point you're making, unless you're saying that would be a more grokable way to state the problem?

deleted 6y ago
[deleted]
qu4z-2 6y ago
It makes a difference how you got there, assuming in the second one that he reliably/intentionally never reveals the grand prize. Imagine you have two identical looking bags of marbles, one contains two red marbles, and one contains one red and one black. You reach into a random one and draw a marble. It is red. If you picked the marble randomly, you now know you're more likely to be holding the bag that contained two red marbles than the mixed bag. The red marble you randomly drew provides bayesian evidence for which bag you're holding. Marble 1 2 Bag 2R R R BR B R Out of the three possible equally likely worlds (given that you know you didn't roll into BR-1), two of them have you holding bag 2R, aka a 2/3rds chance. If, on the other hand, you use a red-marble-picking robot, you get: Marble 1 2 Bag 2R R R BR R R Now you have four possible worlds and are equally likely to be holding either bag, ie 1/2. So, if our friend Monty opens a random door and you only look at simulations where it reveals a goat... There is a 1/3rd chance you initially get the car, but if you do he's guaranteed to reveal a goat. So 1/3rd of games, you have a car. There's a 2/3rd chance you initially get a goat, but then he has only a 50/50 chance of revealing the goat. So half of the 2/3rds of games, the game is discard as invalid, and the other half of 2/3rds (aka 1/3) you have a goat. As such, the potential outcomes are split 1:1:1 between car:goat:invalid. Removing the invalid cases, it's 1:1 car:goat, despite the only 1/3rd chance of you choosing the car initially. tl;dr: If he's randomly opening doors and you discard any where it's the car, those cases are entirely ones where you didn't initially choose the car, which introduces some bias. If he's opening doors deliberately, no cases are thrown out since it's always possible to reveal a goat, maintaining the original 1:2 probability ratio. EDIT: formatting.

Tomminn 6y ago

You only remove games as moot when Monty reveals a car, which can only happen when you haven't initially selected the car. Therefore, no games are removed as moot when you have initially selected a car, and 50% of games are removed as moot when you have initially selected a goat. So it ends in a coin toss whether you switch or not. The 2/3 to 1/3 split only happens when Monty removes a goat using privileged information.