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Consider it framed this way: You pick door #3. The host says they're feeling generous so he is going to give you a free door. He blindly reaches into a bag wit

Consider it framed this way: You pick door #3. The host says they're feeling generous so he is going to give you a free door. He blindly reaches into a bag with three marbles labeled 1,2, and 3. He pulls out the marble labeled 2. He opens door 2 and there's a goat behind it. Do you now switch to door 1? The answer is that it doesn't matter, it's 50/50.

jjnoakes 3y ago
Right, in the world where the door he already opened was done randomly (i.e. it could have been a goat or it could have been a car, and it could have been the door you also chose), then if he happened to both (a) open one of the doors you didn't pick, and (b) he opened a goat door, then switching doesn't matter. But that's only one branch of the probability tree, and if you investigate all of them (assuming he chooses a door at random equally, might open yours, and might open a car), then it's not 50/50, it's 33/66 I think. Perhaps you just weren't clear enough that you think the 50/50 is limited to only a subset of the possible outcomes in your scenario.
- kgwgk 3y ago
  What would "switching" even mean if the door that was opened was the door that you had picked and the car was there? What would be the point of switching to the remaining door if the door that was opened was not the one you picked and the car was there? (The probability of getting the car would be zero either way.) You're right that the subspace of scenarios where (a) and (b) happen is just part of the full space of things that could have happened - but it's the only part relevant for the problem. (We're explicitely told that (a) and (b) are true!)
  - jjnoakes 3y ago
    That's the point though, no? The only way to justify the 50/50 claim made above that I can see is that the host must choose the door to open at random from among all 3 doors. So scenarios do arise where the host opens your door and it is the car, or that the host opens some other unchosen door and it is the car. If you don't allow for those possibilities, then the host really isn't making their choice with equal probability. The 50/50 scenario doesn't make sense.
    - kgwgk 3y ago
      > If you don't allow for those possibilities, then the host really isn't making their choice with equal probability. I don't understand your point. The host can make the choice with equal probability and all those possibilities were initially allowed. Once that a choice was made some possibilities are obviously no longer allowed. > The 50/50 scenario doesn't make sense. This is the original problem: 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. [It's implicit that his knowledge is used to pick a goat with certainty.] He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? This is one possible 50/50 scenario: You pick a door, say #1, and the host picks a number between 1 and 3 from a hat, say #3, and opens that door, 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? The answer in the first variant is "yes, switching from #1 to #2 doubles my probability of winning". The answer in the second variant is "it doesn't matter, the probability of winning is the same". The scenarios that didn't happen are irrelevant. We are not been asked to commit to a strategy in advance. We are asked if we want to switch from door #1 to door #2 after we learn that door #3 was not the good one. That depends only the scenarios that still remain possible and their probabilities. A much simpler problem to stress that the question is whether we want to change conditional on what we know - not all the things that were possible a priory: You're given the choice between two evelopes. One contains $1 and the other $10. You open an envelope and find $1. Is it to your advantage to switch to the other envelope? The answer is obviously yes. If you found $10 the answer would obviously be no. But if you have to commit to switching or not before you see what was your first choice the answer would be "it doesn't change anything". Your answer to the question is conditional on what may have happened. You don't have to "allow for other possibilities" that could have happened but you know already that didn’t.
      - jjnoakes 3y ago
        The original situation covers all scenarios. Any choice of door you make and all possible doors the host might open. The odds are the same every time. The proposed 50/50 scenario only makes sense for some possible situations - those where the host draws a door at random AND the door isn't the same one I chose AND the door opened isn't the car... once you put that situation on equal footing with the original one (all possible doors chosen and all possible doors opened) then it isn't 50/50 any more. Some of those situations are 100% wins (I chose door 3, host opens door 3, car is behind door 3) and some are 100% losses (I chose door 2, host opens door 1, car is behind door 1). Once you look at all outcomes, 50/50 doesn't apply. It becomes 33/66. If your point is that you can cherry-pick a subset of outcomes and get 50/50 odds then I can't really argue, but I'm also afraid I don't how that's useful.
- cman1444 3y ago
  Sure it's only one branch of the probability tree of the game as a whole. But the way the problem is frequently framed is this (I acknowledge that the OP does briefly make the important distinction though): "The game has three doors, two with goats and one with a car. You pick one door. The host selects another for free and shows you a goat behind it. Do you want to switch doors?" My point is that, given this scenario, to make an informed decision you need to ask the host: "Did you know where the goats and cars were? And did you open the goat with certainty, or did you do so at random?" The hosts answer to this question then determines whether you should switch to gain a 33 percent advantage or that it does not matter.

jjnoakes 3y ago

Right, in the world where the door he already opened was done randomly (i.e. it could have been a goat or it could have been a car, and it could have been the door you also chose), then if he happened to both (a) open one of the doors you didn't pick, and (b) he opened a goat door, then switching doesn't matter. But that's only one branch of the probability tree, and if you investigate all of them (assuming he chooses a door at random equally, might open yours, and might open a car), then it's not 50/50, it's 33/66 I think. Perhaps you just weren't clear enough that you think the 50/50 is limited to only a subset of the possible outcomes in your scenario.

kgwgk 3y ago
What would "switching" even mean if the door that was opened was the door that you had picked and the car was there? What would be the point of switching to the remaining door if the door that was opened was not the one you picked and the car was there? (The probability of getting the car would be zero either way.) You're right that the subspace of scenarios where (a) and (b) happen is just part of the full space of things that could have happened - but it's the only part relevant for the problem. (We're explicitely told that (a) and (b) are true!)
- jjnoakes 3y ago
  That's the point though, no? The only way to justify the 50/50 claim made above that I can see is that the host must choose the door to open at random from among all 3 doors. So scenarios do arise where the host opens your door and it is the car, or that the host opens some other unchosen door and it is the car. If you don't allow for those possibilities, then the host really isn't making their choice with equal probability. The 50/50 scenario doesn't make sense.
  - kgwgk 3y ago
    > If you don't allow for those possibilities, then the host really isn't making their choice with equal probability. I don't understand your point. The host can make the choice with equal probability and all those possibilities were initially allowed. Once that a choice was made some possibilities are obviously no longer allowed. > The 50/50 scenario doesn't make sense. This is the original problem: 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. [It's implicit that his knowledge is used to pick a goat with certainty.] He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? This is one possible 50/50 scenario: You pick a door, say #1, and the host picks a number between 1 and 3 from a hat, say #3, and opens that door, 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? The answer in the first variant is "yes, switching from #1 to #2 doubles my probability of winning". The answer in the second variant is "it doesn't matter, the probability of winning is the same". The scenarios that didn't happen are irrelevant. We are not been asked to commit to a strategy in advance. We are asked if we want to switch from door #1 to door #2 after we learn that door #3 was not the good one. That depends only the scenarios that still remain possible and their probabilities. A much simpler problem to stress that the question is whether we want to change conditional on what we know - not all the things that were possible a priory: You're given the choice between two evelopes. One contains $1 and the other $10. You open an envelope and find $1. Is it to your advantage to switch to the other envelope? The answer is obviously yes. If you found $10 the answer would obviously be no. But if you have to commit to switching or not before you see what was your first choice the answer would be "it doesn't change anything". Your answer to the question is conditional on what may have happened. You don't have to "allow for other possibilities" that could have happened but you know already that didn’t.
    - jjnoakes 3y ago
      The original situation covers all scenarios. Any choice of door you make and all possible doors the host might open. The odds are the same every time. The proposed 50/50 scenario only makes sense for some possible situations - those where the host draws a door at random AND the door isn't the same one I chose AND the door opened isn't the car... once you put that situation on equal footing with the original one (all possible doors chosen and all possible doors opened) then it isn't 50/50 any more. Some of those situations are 100% wins (I chose door 3, host opens door 3, car is behind door 3) and some are 100% losses (I chose door 2, host opens door 1, car is behind door 1). Once you look at all outcomes, 50/50 doesn't apply. It becomes 33/66. If your point is that you can cherry-pick a subset of outcomes and get 50/50 odds then I can't really argue, but I'm also afraid I don't how that's useful.
cman1444 3y ago
Sure it's only one branch of the probability tree of the game as a whole. But the way the problem is frequently framed is this (I acknowledge that the OP does briefly make the important distinction though): "The game has three doors, two with goats and one with a car. You pick one door. The host selects another for free and shows you a goat behind it. Do you want to switch doors?" My point is that, given this scenario, to make an informed decision you need to ask the host: "Did you know where the goats and cars were? And did you open the goat with certainty, or did you do so at random?" The hosts answer to this question then determines whether you should switch to gain a 33 percent advantage or that it does not matter.