22 votes

The Monty Hall problem

32 comments

  1. [13]
    ali
    Link
    I love this problem because at first it’s always hard to make someone believe that switching helps, but after having seen it so many times I find it funny that it needs so much convincing. One way...

    I love this problem because at first it’s always hard to make someone believe that switching helps, but after having seen it so many times I find it funny that it needs so much convincing.
    One way to help with the understanding is to imagine more than 3 doors. Let’s say 100 doors. You pick one of them and the host opens all except for the one you have and one other. Would you switch now?

    14 votes
    1. [6]
      aphoenix
      Link Parent
      I love this problem, and I love this explanation. On more than one occasion, I've had someone still refuse to believe that there was a reason to switch, but then this one minor variant was enough...

      I love this problem, and I love this explanation. On more than one occasion, I've had someone still refuse to believe that there was a reason to switch, but then this one minor variant was enough to convince them even though it is functionally the same.

      "Would you rather hve the one door you picked, or all the doors you did not pick?"

      What is hidden is that, functionally, you're giving the person all the doors they don't pick if they switch. That has made the lightbulb come on for some people.

      14 votes
      1. [4]
        rogue_cricket
        (edited )
        Link Parent
        My younger brother once called me at 11PM, drunk, to get me to explain it to all of his equally drunk friends (I have a degree in mathematics so he wanted an "authority"). Apparently they were all...

        My younger brother once called me at 11PM, drunk, to get me to explain it to all of his equally drunk friends (I have a degree in mathematics so he wanted an "authority"). Apparently they were all fighting about it and he couldn't figure out how to express to them that switching was the best choice. I used the 99 door explanation and it went over really well even with a bunch of rowdy drunk folks. It's actually a very fond memory of mine!

        13 votes
        1. [3]
          aphoenix
          Link Parent
          Are you actually me? I've also explained this to my brother and his friends, also while they were drunk, also late at night, because I also have a math degree. I also used this explanation.

          Are you actually me?

          I've also explained this to my brother and his friends, also while they were drunk, also late at night, because I also have a math degree. I also used this explanation.

          10 votes
          1. [2]
            blitz
            Link Parent
            Next challenge: convince your drunk friends that 0.9999... = 1

            Next challenge: convince your drunk friends that 0.9999... = 1

            4 votes
            1. aphoenix
              Link Parent
              I'm having flashbacks to bad parties in university.

              I'm having flashbacks to bad parties in university.

              1 vote
      2. Akir
        Link Parent
        I kind of love this 'paradox' because after you understand how it works you'll be amazed at how simple the problem actually was to begin with. It's one of the reasons why I like the design of Deal...

        I kind of love this 'paradox' because after you understand how it works you'll be amazed at how simple the problem actually was to begin with. It's one of the reasons why I like the design of Deal Or No Deal. It's full of these little probability problems that makes it seem like you can win the big prize but in reality that chance is microscopic.

        7 votes
    2. [6]
      nothis
      Link Parent
      For some reason, this doesn't help me, at all. It's still just 2 doors. I'm pretty convinced this is a psychology problem, not a math problem.

      One way to help with the understanding is to imagine more than 3 doors. Let’s say 100 doors. You pick one of them and the host opens all except for the one you have and one other. Would you switch now?

      For some reason, this doesn't help me, at all. It's still just 2 doors.

      I'm pretty convinced this is a psychology problem, not a math problem.

      4 votes
      1. [3]
        whbboyd
        Link Parent
        I have an extremely clear visualization in my head when I use that example, which maybe helps to illustrate. To start, let's say there are ten doors (to keep my table-art at a reasonable width): 1...

        I have an extremely clear visualization in my head when I use that example, which maybe helps to illustrate.

        To start, let's say there are ten doors (to keep my table-art at a reasonable width):

        1 2 3 4 5 6 7 8 9 10
        . . . . . . . . . .

        Arbitrarily, you select door 2.

        1 2 3 4 5 6 7 8 9 10
        . * . . . . . . . .

        Then Monty opens eight other doors, revealing eight goats.

        1 2 3 4 5 6 7 8 9 10
        g * g g g . g g g g

        …Why didn't Monty open door 6?

        Various mathematical explanations have been done to death, but the mental image of that array of open doors with one left closed I find really evocative of the reasoning: Monty knows which door has the car behind it, and—constrained by the rules of the game—he's showing you that door by opening all the other ones.

        13 votes
        1. [2]
          nothis
          Link Parent
          I get it, mathematically, but it's still weird to me that the prize never changes position yet my choice matters. The reason I suspect psychology is that Monty knowing which door has the car...

          I get it, mathematically, but it's still weird to me that the prize never changes position yet my choice matters. The reason I suspect psychology is that Monty knowing which door has the car behind it isn't really something that's easy for me to intuitively remember, yet it seems to be important for the whole game to work. On the other hand, I don't see the scenario changing with him not knowing, if you just accept the possibility of him having accidentally revealed the winning door (and that simply hasn't happened).

          9 votes
      2. ali
        Link Parent
        It’s 2 doors, but when you made your first pick you had a 1/100 chance of getting the right one. Now 98 wrong choices were removed. If you switch now you can make your choice having so many wrong...

        It’s 2 doors, but when you made your first pick you had a 1/100 chance of getting the right one. Now 98 wrong choices were removed. If you switch now you can make your choice having so many wrong choices removed.

        3 votes
  2. [9]
    Sand
    Link
    I didn't understand this until I realized that the host knows what's behind the doors. Some people omit that part.

    I didn't understand this until I realized that the host knows what's behind the doors. Some people omit that part.

    12 votes
    1. [2]
      Pistos
      Link Parent
      I've read that the probabilities are different depending on whether or not that is the case.

      I've read that the probabilities are different depending on whether or not that is the case.

      4 votes
      1. stu2b50
        Link Parent
        The key part there is that the host always opens a door, and can't open a door where the car is behind it. In which case you can't win. The thought experiment because pretty derpy if they don't...

        The key part there is that the host always opens a door, and can't open a door where the car is behind it. In which case you can't win.

        The thought experiment because pretty derpy if they don't know what's behind the doors.

        8 votes
    2. [6]
      nothis
      Link Parent
      I don't think that's quite it, though. The example should also work with a completely random door being opened. The case of the host opening the door with the car behind isn't relevant, we're...

      I don't think that's quite it, though. The example should also work with a completely random door being opened. The case of the host opening the door with the car behind isn't relevant, we're dealing with the case where there isn't a car in the opened door.

      1. [4]
        aphoenix
        Link Parent
        Actually, the person you responded to is entirely right. The host does explicitly know and will not open a door with a car behind it. The opening of the door is absolutely not random. If it were...

        Actually, the person you responded to is entirely right.

        The host does explicitly know and will not open a door with a car behind it. The opening of the door is absolutely not random. If it were random, the probability would be different.

        I've always liked this slightly simpler version:

        • there are 3 doors
        • there is nothing behind 2 doors
        • there is a prize behind one door

        You get to pick a door, and then the host says "do you want to switch and get everything behind the 2 other doors, or do you want to keep your one door?"

        Everyone understands that they should take what is behind the 2 doors.

        The critical piece of understanding is that I have described the same puzzle.

        It's even more obvious when you make it more extreme:

        • there are 99,999 doors with nothing
        • there is one door with a prize
        • you may pick one door
        • the host offers to let you switch to 99,999 other doors

        You will always take the switch.

        Again, this is a description of the same puzzle, because the host has knowledge of where the prize is.

        @Sand see this comment for explanation.

        18 votes
        1. [3]
          nothis
          Link Parent
          But the moment you end up with a situation where he shows you one of the other doors is empty, it's the same again, no? Even if he didn't know and opened the empty one "by accident"? The only...

          But the moment you end up with a situation where he shows you one of the other doors is empty, it's the same again, no? Even if he didn't know and opened the empty one "by accident"?

          The only situation in which I can imagine the probability changing is if he took away a door, not showing you whether or not it contained the prize. There would be a chance of it having contained the prize and your winning chances would be reduced to 0. Then switching would indeed not matter because that is one of the outcomes.

          1 vote
          1. aphoenix
            (edited )
            Link Parent
            The key is to understand that there is no difference between "you get to keep everything behind all the other doors" and "the host opens a door and shows you that it is empty". That's the reason...

            The key is to understand that there is no difference between "you get to keep everything behind all the other doors" and "the host opens a door and shows you that it is empty". That's the reason that I always take out the goat when discussing this - the goat is a distraction.

            No matter what happens, you have a 1/3 chance of having picked correctly (with 3 doors). That chance does not change when you acquire new knowledge. There is no way to change your original chance of selecting correctly, since that is done in the past, and the past cannot be changed.

            It works like this, and please read carefully because I'm actually doing something slightly different than the classic Monty Hall:

            • There exists doors A, B, C. Each have a 1/3 chance of containing the prize.
            • you choose A. You have a 1/3 chance of having chosen correctly.
            • (B, C) has a 2/3 of having the prize.
            • The host allows that you may keep what's behind door A, or you can switch and take the prize if it is behind B or C.
            • Most people will choose "B and C" instead of "just A"

            I have written something slightly different from the classic example (the host opens one empty door) but this is actually what is happening in the classic problem. The host is doing a bit of performative trickery to make you think that something he does now can change a possibility that occurred in the past but that isn't possible. You cannot retroactively change the odds of your decision being correct, and all the performative trickery in the world cannot change statistics.

            If you're more into "seeing for yourself" this is something you can show empirically. Sit with a friend and run the situation 100 times, and you'll see that the majority favours switching.

            6 votes
          2. skybrian
            Link Parent
            There are 3 possible games depending on where the the car is. If the host opens a door randomly then there is 1/3 chance he shows the car. Since he didn't, one of the possible games has been...

            There are 3 possible games depending on where the the car is. If the host opens a door randomly then there is 1/3 chance he shows the car. Since he didn't, one of the possible games has been eliminated and the probability for the remaining doors is 1/2.

            But if the host knows where the car is and avoids it, he would never terminate any of the 3 games. The car could have been behind any door and you would still be playing the game. It seems the probability that it's behind the door you picked only changes if there was a chance you'd be out of the game by now.

            1 vote
      2. Sand
        Link Parent
        ...then I don't understand it. Isn't it just random in that case?

        ...then I don't understand it. Isn't it just random in that case?

        2 votes
  3. Pistos
    Link
    I was one of the people that thought staying and switching had equal odds of winning. I somewhat understood some of the proofs given, but was fully convinced after implementing and executing a...

    I was one of the people that thought staying and switching had equal odds of winning. I somewhat understood some of the proofs given, but was fully convinced after implementing and executing a simulation myself. The wonder of the probability, math and logic of the problem is somewhat interesting, but what's perhaps more fascinating is the psychology aspect, how there is such a division of opinion among people, and how the human mind can confidently believe something which is provably incorrect.

    7 votes
  4. Eric_the_Cerise
    Link
    I've heard of it before, but this is my first time thinking it through. Here's my explanation, only 3 doors required. When you first choose, there is a 1 in 3 chance that you have chosen the car....

    I've heard of it before, but this is my first time thinking it through. Here's my explanation, only 3 doors required.

    When you first choose, there is a 1 in 3 chance that you have chosen the car. Switching guarantees you a goat 1time in 3.

    However, there is a 2 in 3 chance that you've chosen a goat, after which, Monty chooses the other goat. No goats left. Switching guarantees you the car, 2times out of 3.

    2 votes
  5. [7]
    Wes
    Link
    I thought bald Wikipedia links were a no-go on Tildes? https://tildes.net/~tildes/9yl/suggestion_wikipedia

    I thought bald Wikipedia links were a no-go on Tildes?

    https://tildes.net/~tildes/9yl/suggestion_wikipedia

    1 vote
    1. [2]
      Pistos
      Link Parent
      I didn't know about this past history. If I did, though, I would have just posted a URL from some other domain, because it's the topic itself that I wanted to share, not specifically Wikipedia's...

      I didn't know about this past history. If I did, though, I would have just posted a URL from some other domain, because it's the topic itself that I wanted to share, not specifically Wikipedia's treatment of it. And I did provide a top-level comment, which hopefully reduces its qualification as a low-effort post.

      But thanks for bringing this to everyone's attention.

      3 votes
      1. Wes
        Link Parent
        I think your starter comment itself might have been an interesting submission, with the link to Wikipedia being a resource added at the end.

        I think your starter comment itself might have been an interesting submission, with the link to Wikipedia being a resource added at the end.

        3 votes
    2. cfabbro
      Link Parent
      In some people's opinions that may be the case, but the submission history and votes says otherwise: https://tildes.net/?tag=wikipedia

      In some people's opinions that may be the case, but the submission history and votes says otherwise:
      https://tildes.net/?tag=wikipedia

      2 votes
    3. [3]
      jlj
      Link Parent
      I don't think I have enough experience on forums like this one to add anything insightful to that particular discussion. I do give a lot of credence to Deimos' take on it, though. It's tough,...

      I don't think I have enough experience on forums like this one to add anything insightful to that particular discussion. I do give a lot of credence to Deimos' take on it, though.

      It's tough, though, because I've enjoyed this thread so much. :-) But I'm coming around to the idea that it might be the exception.

      1 vote
      1. skybrian
        Link Parent
        It seems like on a link-sharing site we are often saying "<link>, discuss." But there is a box where you can say something more and I usually add a quote of the part I thought was interesting.

        It seems like on a link-sharing site we are often saying "<link>, discuss." But there is a box where you can say something more and I usually add a quote of the part I thought was interesting.

        3 votes
      2. moocow1452
        Link Parent
        I totally want to post Giraffes, discuss in ~talk now.

        I totally want to post Giraffes, discuss in ~talk now.