Please help solving this Year 5 probability problem

srr on 27/12/2022 - 23:19

My boy got this problem in Year 5:

Ben and Jenny roll a dice once. The first one to roll a five wins. What are the probability of winning?

The teacher gave an answer 1/6 for both players. I mentioned that whoever starts first has an upper hand as the second player doesn't have a chance to roll at all.

I asked Chat GPT, and the AI was confused at first. However, after a few pointers came up with a good reasoning when Ben starts first:

In this game, the players take turns rolling a dice with 6 sides. The first player to roll a 5 wins.

There are 3 possible outcomes in this game: Ben winning, Jenny winning, or a draw.

The probability of Ben winning is 1/6 on his first roll.

The probability of Jenny winning is the probability that Ben does not roll a 5 on his first roll, which is 5/6, multiplied by the probability that Jenny rolls a 5 on her first roll, which is also 1/6. Therefore, the probability of Jenny winning is (5/6) * (1/6) = 5/36.

The probability of a draw is the probability that neither player wins, which is the probability that both players roll a number other than 5. This can be calculated as (5/6) * (5/6) = 25/36.

As a bonus, what are the probabilities of winning if Ben and Jenny keep rolling until someone wins?

Poll Options

  • 70
    1/6 and 5/36
  • 86
    1/6 and 1/6
  • 182
    the problem is too complex for Year 5
  • 7
    your answer
Education & Work

Comments

  • Hybroid on 27/12/2022 - 23:25
    +62

    You have made the problem too complex for Year 5. Assume who goes first isn't relevant so it's 1/6.

    After all, all probability outcomes are 50-50. Either an outcome happens, or it doesn't.

    • srr on 27/12/2022 - 23:28
      +1

      But how you select who goes first? If it was a casino game, would you play it if the dealer always goes first?

      • Hybroid on 27/12/2022 - 23:37
        +49

        Your thinking isn't wrong and question could have been clearer, but you're reading too much into a Year 5 problem. The question (and their answer) is meant to be in isolation so not relevant who goes first at this level.

        Simply what's the probability of any person rolling a five. They add 2 people to outline that it's the same probability for both of them.

        In more advanced levels, they might indicate Ben goes first and ask to calculate probability in that case, which would be akin to your thinking.

        • srr on 27/12/2022 - 23:49
          +7

          That's exactly my concern with this problem. In my books there's only one correct answer regardless if you're a Year 5 student or have a PhD degree. I tried to help my son and taught him how to use sample space, but then realised it's a little bit too early for him. On the other hand just saying "it's 1/6" can lead to great misunderstanding of probability theory.

          • Hybroid on 28/12/2022 - 00:07
            +47

            @srr: It’s not a different answer. It’s a different context.

            You’re not doing conditional probability at Year 5 level.

            • Scrooge McDuck on 28/12/2022 - 12:14
              +9

              @Hybroid: Context doesn't matter to mathematics. There should be an indisputable answer.

              • freefall101 on 28/12/2022 - 13:31
                +6

                @Scrooge McDuck: Ignoring that there is an indisputable answer (that Ben rolls first is assumed by OP, it's not in the question), context does matter to mathematics. Particularly when applied to physics. If you throw a ball, the size of the ball, the gravity, the assumption that physics is constant throughout the universe, the speed at which you're traveling, the observer of the ball all matter.

                But if you gave a kid a question of measure the arc of a ball when thrown, you wouldn't be asking the 11 year old to take into account all of the above, because it's just ridiculous. Same as with throwing a dice, just assume they both throw at the same time and the answer makes perfect sense. And it sounds like the kid didn't need a parent overthinking the problem for them when asking for help.

                I studied physics until year 12 (and gave it up because it's too hard and there's more money elsewhere), most of what I learned at aged 12-15 was actually wrong because it used Newtonian physics. But a baseline understanding of physics is more important than the "right" answer at that stage.

                • Scrooge McDuck on 28/12/2022 - 15:23
                  +2

                  @freefall101: You made a claim about mathematics and then proceeded to give reasoning about physics. Mathematics and physics are not equivalent.

                  I don't wish to get into a debate about the definitions of scientific fields. My preliminary web search indicated that there is murkiness there. However, roughly, physics is the study of nature particularly at the largest and smallest scales of size. Whereas, mathematics is more focused on abstract logic. Whether or not you regard mathematics as a subset of physics, physics is not a subset of mathematics. Physics is a more general study of the universe than mathematics.

                  The smallest scales of physics are known to be much more complex than what an average person perceives of nature. Physics aims to provide models of nature which are accurate at certain scales.

                  Newtonian physics (classical mechanics) isn't wrong. It is an accurate model for a range of scales. Similarly, general relativity and quantum mechanics may be found to be inaccurate at more extreme scales, but that wouldn't make them wrong either.

                  The central issue you raise is what the teacher expects the correct answer to be. The solution to that involves social sciences. To accurately model nature we need to consider everything which is relevant. An example of this would be professional engineering. To approach a simplified hypothetical in a physics question; we know that we are being assessed on certain concepts, so it's reasonable to assume that everything which isn't referenced in the question isn't relevant. These assumptions apply to the physics, not the mathematics of computing the question.

                  • freefall101 on 28/12/2022 - 19:37
                    +3

                    @Scrooge McDuck: You missed the point. Measuring an arc is purely mathematics, the conditions surrounding it are the physics.

                    Likewise the conditions of rolling a dice aren’t mathematics. The answer of 1/6 for each is correct until it was decided Ben rolled first and they had to take turns. That’s why there can be more than one answer.

                    You’ll note nowhere did op say it was a 6 sided dice, because it’s reasonable to assume. This question as myriad answers though because you could assume any number of sides on the dice. Much like you can assume someone rolled first.

                    And Newtonian physics is wrong. A good estimate is still wrong.

                    • srr on 28/12/2022 - 23:28
                      +1

                      @freefall101:

                      The answer of 1/6 for each is correct until it was decided Ben rolled first and they had to take turns

                      Discussed here - under no circumstances the probability of winning is 1/6 for both players.

                      https://www.ozbargain.com.au/comment/13168980/redir

                      • freefall101 on 30/12/2022 - 00:53
                        +3

                        @srr: Discussed further below, all you need to do is accept both rolling 5 as both winning.

                        The question is badly worded, but it clearly says “Ben and Jenny roll a dice once”, not “Ben rolls a dice and if he wins Jenny isn’t allowed to roll” or implies one had to roll first, you inserted that to get a different answer.

                        The teacher wrote a terrible question and you made terrible assumptions about it. The real question is though, what did you tell your kid? Are they now confused as anything and understand nothing about probability or did you make sure they at least understand that rolling a dice gives you a 1/6 chance of any given result? Because that’s the important bit here.

                    • AnophthalmiaCervidae on 29/12/2022 - 10:35
                      +4

                      @freefall101: If Ben was on Einstein's very fast train and Jenny was on the platform and they both rolled the dice at the moment Ben and Jenny passed each other an observer on the platform and an observer on the train might see a different person as having rolled first therefore the winner would depend on the observers frame of reference.
                      But I'm sure special relativity and time dilation were not intended to be part of the year 5 maths problem either.

          • markathome on 28/12/2022 - 01:24
            -1

            @srr:

            That's exactly my concern with this problem. In my books there's only one correct answer regardless if you're a Year 5 student or have a PhD degree. I tried to help my son and taught him how to use sample space, but then realised it's a little bit too early for him. On the other hand just saying "it's 1/6" can lead to great misunderstanding of probability theory.

            Well said.

          • Scrooge McDuck on 28/12/2022 - 13:55
            +3

            @srr:

            Ben and Jenny roll a dice once. The first one to roll a five wins. What are the probability of winning?

            If this truly was the question it contains a redundancy and an ambiguity.

            If Ben and Jenny roll a dice once they are a group and roll a die one time as a group. Therefore "the first one" is redundant since they, as a group, are the only entity mentioned — they are both first and last.

            "What are the probability of winning?" Contains a grammatical error which introduces an ambiguity. Are is plural whereas probability is singular. So now there is doubt about whether Ben and Jenny are a group or individual contestants.

            If that truly was the way the question was written, whoever wrote it was deficient. If not, you should make the effort to report the question correctly so as not to waste the time and effort of the responders.

            • srr on 28/12/2022 - 23:35

              @Scrooge McDuck: Unfortunately I don't have exact wording. Could you please suggest wording that would be suitable for Year 5 students, containing a game of dice with two players, where "the first one who rolls a 5 wins"? Alternatively you can help by solving all possible interpretations of the question.

          • [Deactivated] on 29/12/2022 - 02:02
            +1

            @srr: The teacher obviously means both people roll their dice at the same time. But the question is poorly written.

            1. First, Ben & Jenny don't "roll a dice once." The teacher has used the word "a" and also said "1 in 6" is the answer. Which means each person has only ONE… DIE… not "dice."

            2. Second, they have a 1 in 6 chance of rolling a 5. So both will probably have to roll more than once to get a 5, because they have 5 in 6 chances of NOT rolling a 5. But the question says they only roll ONCE. So after one roll the game is over and probably NEITHER wins.

            This is probably why the teacher then says: "The first one to roll a five wins." Because they are now hinting more than one throw has to occur for someone to roll a 5.

            In other words, what the teacher is trying to say (rather poorly) is:

            "Ben and Jenny each have a die. The first person to roll a five wins. What is their probability of winning?"

            Or…

            "Ben and Jenny each have a die they roll at the same time. The first person to roll a five wins. What is their probability of winning?"

            The fact the teacher said each has 1 in 6 chance, proves this what the wording of their question should have been.

            • chriise on 29/12/2022 - 02:39

              @[Deactivated]: If they both roll at the same time and they both roll a 5, do they both win? This is how I read the question, and I assumed that outcome would be a draw.

              That leads the answer to be 1/6 x 5/6 = 5/36.

              As in order to win, you require a 5 and your opponent to not roll a 5.

              • [Deactivated] on 29/12/2022 - 08:11

                @chriise: Yeah it's weird… teacher could mean "die" when she said "dice" but there's only one and they take turns. The kids would just have to cover every option I guess. i.e. Write an answer for every possibility, then put their hand up and ask, "Did you mean a, b, c, etc?" then pull the page that applies. (I remember in school I did this once and submitted all the answers and got marked DOWN. So it's best to pull the one page that applies, once the child knows what exactly the teacher is on about.)

            • larndis on 29/12/2022 - 10:32
              +1

              @[Deactivated]: @Faulty P xel we don't know if the teacher used any of those words, we only have OP's paraphrased recollection of the question. No point picking apart the wording.

          • doobes on 30/12/2022 - 07:36

            @srr: Nowhere does it say the same dice. They each have a dice.

        • hawkshead on 29/12/2022 - 11:25

          rules state there can be a draw i think in the edit, which removes the first or 2nd roller.

      • monkeyfood on 28/12/2022 - 13:23

        all the casino games are advantage to the bank, you either play them all knowing you will lose or don't play any

      • CalmLemons on 29/12/2022 - 00:12

        Nice try ChatGPT. We know it's you, AI.

      • Finde on 29/12/2022 - 23:14

        Where in the question does it say one goes first and they don't roll the dice at the same time?

      • TheBean on 30/12/2022 - 22:30

        They both roll at the same time?

    • SteveD on 28/12/2022 - 13:25
      +5

      @Hybroid you are absolutely right. OP is making it too complicated for a Year 5 problem.
      There is no question that answer is 1/6 for both players.
      But the vital infomation that is missing from the question is that there were 2 dice and both players rolled it simultaneously. End of discussion.

      • belongsinforums on 28/12/2022 - 17:16
        +2

        If there were only one die, the word die would have been used.

        • srr on 28/12/2022 - 23:36

          *die" discussed here https://www.ozbargain.com.au/comment/13170820/redir

          • belongsinforums on 29/12/2022 - 01:30

            @srr: Sry I red it comenT n i C now I woz rong about english

        • [Deactivated] on 29/12/2022 - 02:05

          No, because the teacher said "a dice" - the answer is: the teacher is illiterate. It's either two people "have dice" or "Ben and Jenny each have a die."

          • StalkingIbis on 29/12/2022 - 08:08

            @[Deactivated]: That hardly makes them illiterate :s

            • [Deactivated] on 29/12/2022 - 08:18

              @StalkingIbis: They're supposed to be teachers. They need to up their game.

          • doobes on 30/12/2022 - 07:36

            @[Deactivated]: Die and dice can be used for single dice.

          • larndis on 30/12/2022 - 09:51
            +1

            @[Deactivated]: @Faulty P xel no, OP said 'a dice'. We have no idea how the teacher wrote the question, so 'illiterate' is entirely groundless and excessive - unless you meant OP?

    • spackbace on 28/12/2022 - 14:33
      +8

      I still remember this, I had a question in year 8 which was along the lines of "if 1 towel takes 20 minutes to dry on the line, how long will 4 towels take to dry?" (yes, I was actually in the upper math class too). I thought about it for a second, answered 20 minutes (after all, the wind doesn't just go to each towel). Teacher marked it wrong, even my parents were dumbfounded

      Fast forward to the parent/teacher interviews and the questioned her on it, and she still couldn't see fault in it!

      So sad sometimes

      • redfox1200 on 28/12/2022 - 14:59

        Was the correct teacher answer 5 minutes or 80 minutes?

        • spackbace on 28/12/2022 - 15:00

          Teacher claimed the 80 minutes was the correct answer

          Again, upper level Math class, year 8, in a pretty decent public school

          • Scrooge McDuck on 28/12/2022 - 15:41
            +11

            @spackbace: If 1 teacher takes 3 years to graduate university, how long will 4 teachers take to graduate?

            This was a problem of communication. The teacher was thinking of the resource of the line being used sequentially, but failed to communicate that. You visualised every clothesline you've seen and assumed they'd be processed in parallel.

      • [Deactivated] on 29/12/2022 - 02:06

        The answer is "unknown" because some towels could now cast shade on the others.

    • ChickenTalon on 28/12/2022 - 22:05
      +3

      After all, all probability outcomes are 50-50.

      Huh?

      All probability outcomes are not 50-50.

      Either an outcome happens, or it doesn't.

      Do you mean all outcomes are binary? No, they’re not. You can have many different outcomes depending on the question.

      In this example you can have three outcomes. Ben win, Jenny win or a draw.

      • Hybroid on 29/12/2022 - 00:06
        -4

        There’s a school of thought that suggests the probability of any of those happening is simply 50-50. Either they do or don’t.

        Likewise probability of rolling a five is 50-50. Either it’s a five or it’s not. But that’s a completely different topic of debate.

        • chriise on 29/12/2022 - 02:43

          school of thought

          School of memes

        • ChickenTalon on 29/12/2022 - 13:04

          There’s a school of thought that suggests the probability of any of those happening is simply 50-50. Either they do or don’t.

          Never heard of that school of thought. But if you phrase the question so that the outcome variable is binary categorical: “what are the odds that Ben wins”. Then you have a binary outcome of Ben winning or not (in the case of not: Jenny wins or a draw).

          Likewise probability of rolling a five is 50-50. Either it’s a five or it’s not. But that’s a completely different topic of debate.

          No. The odds of rolling a five on a regular six sided die is 1 in 6. It’s not “50-50”

    • hawkshead on 29/12/2022 - 11:23

      This is why they need to start teaching preschooler's about the kreb cycle instead of just saying the damned catepillar is hungry hungry!!

    • supersabroso on 29/12/2022 - 13:31

      No, he hasn't, it's just poor writing from whoever made up the question, he's perfectly correct. We should expect better, it's dodgy stuff like this that screws up kid's scores during exams. The proper exams are unlikely to have dumb questions like this, they will be exact and precise, you miss one word and you're wrong.

      They should have said that they both kids get a chance to roll the dice first. That way, they get the same correct answer regardless of whether the student is smart enough to realise that whoever goes first gets an advantage or not.

  • jv on 27/12/2022 - 23:29
    +11

    I mentioned that whoever starts first has an upper hand as the second player doesn't have a chance to roll at all.

    Correct.

    The teacher phrased the question wrong.

    • MrFrugalSpend on 29/12/2022 - 08:34

      Indeed - the teacher phrased the question wrong.
      It should have read something like this:

      "Ben and Jenny each roll a die once. To win (or draw), they need to roll a five. What is the probability of winning (or drawing in the case of both rolling a five) for each person?"

  • random12 on 27/12/2022 - 23:49
    +5

    You get an infinite series

    1 + x+ x^2 + …. = 1/(1-x) for -1 < x < 1

    • markathome on 28/12/2022 - 00:35
      +5

      You get an infinite series

      Correct (and kudos to the folks below who actually solved it) except I think the questioner covered this scenario by stating…

      Ben and Jenny roll a dice once. The first one to roll a five wins.

      Granted it is unusual not to loop the game until there is a winner. But it does make the calculations much easier. Humans like games with outcomes but computers are pretty indifferent on the subject.

      It's impressive Chat GPT gave the correct answer as it is famously bad at quantitative reasoning…

      https://twitter.com/djstrouse/status/1605963129220841473

      But I guess someone answered a very similar question recently on the internet.

      • srr on 28/12/2022 - 01:03

        Re-roll was a bonus question. As I said, ChatGPT needed some teaching, specifically pointing who starts first and that there’s probability of a draw.

      • Chandler on 28/12/2022 - 10:38
        +2

        Question is also phrased poorly.

        Ben and Jenny roll a dice once. - implies one roll only.

        The first one to roll a five wins. - implies multiple rolls until a winner is found.

        Given the context of the second half of that statement, the "once" in the first half could be taken as rolling once each per round, potentially in turns like srr thought.

  • random12 on 27/12/2022 - 23:51
    +4

    First person

    (1/6) + (5/6)^2*(1/6) + (5/6)^4 *(1/6) + …

    = (1/6) [ 1+ (5/6)^2 + (5/6)^4 + … ]

    = (1/6) * (1/( 1- (5/6)^2)) = approx 54% (slight advantage of going first)

    The second person chance is 1 - above = 46%.

    Each term is the previous multipled by (5/6)^2 because both people have to fail to win for there to be the next round.

    Lol, this should not be a year 5 question. This is a year 11 level question.

    • srr on 27/12/2022 - 23:56
      +1

      (1/6) * (1/( 1- (5/6)^2)) = 0.5454545455

      I have an easier solution using sample space:
      probability of Ben winning 1/6, Jenny 5/36, draw 25/36
      however each draw results to re-roll, which in the sample space maps to winning points, contracting the space to 5/36 + 6/36
      first person then has probability of 6/11, while second 5/11
      6/11 = 0.5454545455

      • random12 on 27/12/2022 - 23:58
        +2

        Good intro to conditional probability (lol well outside year 5). Conditional probability is first year university maths.

      • HairyChickens on 28/12/2022 - 00:30

        Is this for the selective high school test lol?

        If yes then damn they are really making it difficult these days

    • Themartinator on 30/12/2022 - 22:53

      They roll the dice once.

  • SpicyPeanut on 27/12/2022 - 23:54

    The correct answer is X Æ A-12

    • iDroid on 28/12/2022 - 23:11
      +1

      Harsh crowd.

  • RandomNinja on 28/12/2022 - 00:01
    +26

    Does the question indicate that Ben and Jenny roll dice consecutively? If not, couldn't Ben and Jenny each roll a die simultaneously, and therefore their individual odds of rolling a 5 is 1/6?

      • timthetoolman on 28/12/2022 - 07:57
        +2

        Why is that? They both at some point happen, and have happened, so have a finish time of the event.
        Why can't those two events happen at exactly the same time?

      • larndis on 28/12/2022 - 08:28
        +13

        Really, you have absolutely no concept of the meaning of the word simultaneously?

      • nobro25 on 28/12/2022 - 11:37
        +1

        Ahhh. Maybe there's two dice and both Ben and Jenny have a dice each to roll for their own outcome. Basically the first person that rolls a 5 wins and if they both roll a 5 then it's a draw and they repeat in the next round.

    • random12 on 28/12/2022 - 00:15
      +1

      The first one to roll a five wins.

    • markathome on 28/12/2022 - 00:25

      If both rolled the dice simultaneously and consecutively. The chance of either winning is (1 - 1/36) / 2 or 35 / 72. The chance of a draw is 1 / 36.

      If both rolled the dice simultaneously once. The chance of either winning (1 / 6) * (5 / 6) = 5 / 36. The chance of a draw is 1 / 36 and the chance of no result is 25 / 36.

      • dsp007 on 28/12/2022 - 00:49

        If both rolled the dice simultaneously once, isn't the chance of either winning 1/6 + 1/6 = 1/3?

        Basically the same as someone rolling a dice twice

        Edit: I see what you did. 5/32 if one wins AND not the other

        • markathome on 28/12/2022 - 01:16
          +1

          Yeah but you are technically correct (provided both winning counts as either winning) and, if I'm being a bit pedantic, it's a bit rich for me to then turn around and argue you're being too pedantic. Perhaps I should have written:

          If both rolled the dice simultaneously once. The chance of first player winning is (1 / 6) * (5 / 6) = 5 / 36. The chance of the other player winning is (1 / 6) * (5 / 6) = 5 / 36. The chance of a draw is 1 / 36 and the chance of no result is 25 / 36.

    • srr on 28/12/2022 - 00:46
      -6

      The problem with no order is that we have to build a model of who rolls dice faster. One asdumption is Ben and Jenny absolutely equal, but if I know anything about people they are not random. People generally predictable, so there will be a bias towards one player. The problem gets even more complex.

      By the way their individual probability of rolling a 5 is 1/6 anyway.

      • Kail on 28/12/2022 - 01:02
        +16

        Why on earth would rolling speed be a factor in a simple dice game?? If they are rolling together they would be doing it in rounds. You know, like when players roll dice to see who goes first at the start of a game, and everyone gets to roll before the results are looked at? You are way overcomplicating things.

        • srr on 28/12/2022 - 23:42

          Why on earth would rolling speed be a factor in a simple dice game??

          If they each have a dice, and roll after a signal, then it's important who will have the outcome first. Because in case of 5,5 the first one wins.

    • AlanJ on 28/12/2022 - 09:12

      That's a good point. When the question says "roll a dice", it is sending two messages: "a" means one, implying a single die, so they have to take turns; "dice" means at least two, so they must be rolling different dice, implying simultaneous rolls. The question baffled me before I ever got to the Maths!

  • SF3 on 28/12/2022 - 00:02
    +1

    My boy got this problem in Year 5

    Professional gambler career path?

  • dsp007 on 28/12/2022 - 00:20
    +10

    Are those the exact words of the question?

    The original question in the year 5 example never specified if they went one after another or at the same time.
    My comprehension skills tells me Ben and Jenny roll a dice each together…

    • srr on 28/12/2022 - 00:30
      -7

      I don’t have the exact wording. Definitely remember “the first who rolls wins”. Do you have exact wording at hand?

      • dsp007 on 28/12/2022 - 00:34
        +6

        I do not have the question but one thing I learnt studying maths is always read the question multiple times before answering as a single word can change the outcome.

        Also in this "scenario", they can both win at the same time so still seems valid thus I think that is the intention of the question. If it was missing those key words, then it's just a poorly worded question with multiple answers as it's open to interpretation

        • srr on 28/12/2022 - 00:48

          The first to roll 5 wins can’t lead to both winning.

          I’m thinking of how to get to 1/6:

          Ben and Jenny each roll a dice but keep result in secret. Then they throw a coin - if tails Jenny opens first, if heads Ben. The first to uncover 5 wins.

          • dsp007 on 28/12/2022 - 00:53
            +5

            @srr: Well it can lead to both winning. If they both roll at the same time and both land on 5, they both technically win (How many times have gold medals been shared at an Olympics when there should only be 1 winner?).

            Could also be if this happens, then it's a draw and they keep going until one person wins. Again an open ended question leads to open ended answers

  • GrueHunter on 28/12/2022 - 00:22
    +29

    Did everybody clap? (And what help are you looking for, exactly? Oh wait, my bad - you just wanted to bang on about that time you spotted a linguistic technicality in a child's homework.)

    Here's the relevant content element for the Australian curriculum for Year 5 mathematics:

    "List outcomes of chance experiments involving equally likely outcomes and represent probabilities of those outcomes using fractions."

    The point of the question - and let's be blunt here, you know perfectly f&^%ing well what the point is - is to illustrate that previous rolls of a die don't influence future rolls. Any time somebody throws that die, they have a 1 in 6 chance of rolling a 5. And that's it, because the question was for a child in Year 5.

    Next time, maybe try "ACK-shu-ally, wHaT iF tHe DiE iS a dOdEcAhEdron?! You didn't specify! snort".

    • srr on 28/12/2022 - 00:26
      -1

      Thank you for your help.

    • p1 ama on 28/12/2022 - 00:54
      -1

      Did everybody clap? (And what help are you looking for, exactly? Oh wait, my bad - you just wanted to bang on about that time you spotted a linguistic technicality in a child's homework.)

      It's actually not a technicality in the way the question is worded, because regardless of how the game is played (see my post below), there is in no circumstance where the probabilities of winning are both 1/6. It does not matter whether they both roll at the same time, or whether they roll sequentially.

      (See below)

      The point of the question - and let's be blunt here, you know perfectly f&^%ing well what the point is - is to illustrate that previous rolls of a die don't influence future rolls.

      As someone who used to be a teacher at various levels of mathematics (including up to university level), and has written exercises, exams, assignments for all different levels of students (though admittedly, not primary school), the question needs to be clear. The "point" of the question is irrelevant, the question is what is written, not what is the "point" of the question. This is why we have other teachers cross-check our questions so that they actually check what's on the paper, not what we meant in our heads.

      Regardless, since the question does not involve multiple rolls by the same individual, the independence of future rolls is actually irrelevant. So you're actually wrong here.

      And that's it, because the question was for a child in Year 5.

      Again, all the more reason for the question to be clear.

      Next time, maybe try "ACK-shu-ally, wHaT iF tHe DiE iS a dOdEcAhEdron?! You didn't specify! snort".

      I'm glad you take joy in being a smart-ass.

      • markathome on 28/12/2022 - 01:12
        +2

        This. I wrote a much more extreme reply. But there's nothing wrong in teaching kids maths is complex. And there's a lot right in teaching them that precision matters.

        Matt rolls a five on a fair six-sided dice. What is the chance his friend Tom then also rolls a five?

        Simple question with 1 / 6 being the answer.

      • srr on 28/12/2022 - 10:10

        Funny you got negative votes. I wonder why ?

        • pegaxs on 28/12/2022 - 15:54
          +1

          Because p1 didn't reinforce the hive mind's collective stupidity.

          Doesn't matter how right you are, there are always those who don't know that think they know or who don't like a right answer that doesn't reaffirm their answer they think they know, so they neg.

          I've lost count of the number of times I have been negged to oblivion over just simply posting a fact with a link to the relevant source (ie: road rules) and it's mostly because people don't like the answer, as if I personally write it, or it goes against what they were taught 30 years ago and don't like being made to look like an idiot for having the wrong answer.

          Basically, you can be as right as you like, but if the neg sheep don't like the answer, right or not, you're gonna get negged.

          p1 is probably being negged because they didn't just say "it's 1/6"…

          • Scrooge McDuck on 28/12/2022 - 16:52

            @pegaxs: Do you derive more utility from posting here than the time and effort you expend on it?

            I ask because I realised a few years ago that I don't. There seems to have been a proliferation of negativity and stupidity on the internet.

            • srr on 28/12/2022 - 23:49

              @Scrooge McDuck:

              There seems to have been a proliferation of negativity and stupidity on the internet.

              That's why I'm not on any social network. When I have a rare urge to discuss something on the internet, OzBargain is my choice. I have a theory that savvy people have a tad bit more critical thinking skills.

            • pegaxs on 29/12/2022 - 21:58
              +2

              @Scrooge McDuck: I come here to be confronted by stupidity, because, in turn, it makes me do more research. I dont do it for any other reason than to be up to my neck in misinformation so as I may educate myself. The collateral damage from that is that a small number of people may also be educated on the facts. Not a lot, but some.

              I have found that some people cannot be educated. There are a hell of a lot of people that like their ignorance bubble and will do anything to protect it. I never want to stop learning or finding out something new or the truth and I dont care how many butt-hurt negs I get from the neg sheep, because I know I researched it and if they hate me because they are too stupid to accept a challenge to their knowledge, then so be it. Negging me isnt going to change facts and facts dont care about the negger's feelings either.

              I guess, at the end of the day, it's a zero sum game. I learn what I need to, but at the same time, I am inundated with idiots who refuse to accept that their could be/are wrong. It balances out, for the moment.

          • p1 ama on 30/12/2022 - 14:58
            +1

            @pegaxs:

            Doesn't matter how right you are, there are always those who don't know that think they know or who don't like a right answer that doesn't reaffirm their answer they think they know, so they neg.

            Just a case of intuition bias - if something is not intuitive, then it must not be correct. It if weren't for the ability to fly around the world, I'm sure we'd have a lot more flat-earthers, for instance.

            The reality is that a lot of people just do not have a very good understanding of probability and statistics. How often do you hear "I put $X on Nadal at $1.20, he's obviously going to win", without understanding that the odds are already implying he has an 83.3% chance of winning.

    • Scrooge McDuck on 28/12/2022 - 16:03

      Are you a teacher?

  • Kail on 28/12/2022 - 00:45
    +9

    The teacher answered for a question involving two players rolling dice at the same time in a series of rounds.
    You answered for a question involving two players taking turns rolling dice, and maintaining the order of those turns.

    Either the teacher worded the question incorrectly, or you read the question incorrectly.
    Without the exact wording, we cannot tell which.

    I suppose the lessons to learn here are, be specific with your wording and be thorough with your reading.

  • p1 ama on 28/12/2022 - 00:46
    +30

    Heya, I taught probability at a university level for several years and was also a researcher in statistics. I can have a crack.

    Ben and Jenny roll a dice once. The first one to roll a five wins. What are the probability of winning?

    You are absolutely right that whoever goes first has an advantage. However, let's consider first the case if both of them roll at the same time.

    In this case, there's a 1/6 chance Ben rolls a 5, a 1/6 chance Jenny rolls a 5, implying that there is a 5/6 chance Ben does not roll a 5 and a 5/6 chance Jenny does not roll a 5.

    Therefore, in this case:

    Pr(Ben Wins) = Pr(Ben 5, Jenny Not 5) = 1/6 * 5/6 = 5 / 36
    Pr(Jenny Wins) = Pr(Jenny 5, Ben Not 5) = 1/6 * 5/6 = 5 / 36
    Pr(Draw due to Both 5) = 1/6 * 1/6 = 1 / 36
    Pr(Draw due to Both not 5) = 5/6 * 5/6 = 25 / 36

    If you add up all four possible cases, then of course, you end up with 36 / 36 = 1 (which is a good double check).

    Okay, now the second scenario where one person goes first. Let's just assume Ben does, but you can swap the names if you wanted Jenny to go first.

    The best way to think about this is a tree. In the first step, Ben will roll, he can either roll a 5 or not 5. Therefore:

    Pr(Ben wins) = Pr(Ben 5) = 1/6
    Pr(Game continues) = Pr(Ben not 5) = 5/6

    Now, under the second branch, Jenny will roll, of course, she can either roll a 5 or not (and if not, it will be a draw):

    Pr(Jenny wins) = Pr(Jenny 5, Ben not 5) = 1/6 * 5/6 = 5/36
    Pr(Draw) = Pr(Jenny not 5, Ben not 5) = 5/6 * 5/6 = 25/36

    Therefore, the probabilities are:
    Ben wins 1/6 (i.e. 6/36), Jenny wins 5/36, and draw 25/36 (which again, adds up to 36/36, voila!)

    As a bonus, what are the probabilities of winning if Ben and Jenny keep rolling until someone wins?

    Under case 1, where they both roll at the same time, and any type of draw means a re-roll, then it will be 50/50, so the probability of either Ben or Jenny winning is 1/2.

    Under case 2, where Ben goes first, then you need to think of it as a tree again.

    So in the first case:

    Pr(Ben 5) = 1/6 -> Ben wins
    Pr(Ben not 5) = 5/6

    Then the second level:

    Pr(Jenny 5, Ben not 5) = 1/6 * 5/6 = 5/36 -> Jenny wins
    Pr(Jenny not 5, Ben not 5) = 5/6 * 5/6 = 25/36

    Then the third level (which will go on if Jenny not 5 and Ben not 5):

    Pr(Ben 5, Jenny not 5, Ben not 5) = 1/6 * 5/6 * 5/6 -> Ben wins
    Pr(Ben not 5, Jenny not 5, Ben not 5) = 5/6 * 5/6 * 5/6

    Then the fourth level (which will go on in the second case of the last level):

    Pr(Jenny 5, Ben not 5, Jenny not 5, Ben not 5) = 1/6 * 5/6 * 5/6 * 5/6 -> Jenny wins
    Pr(Jenny not 5, …) = 5/6 ^ 4

    You can probably see a pattern from here, which is that:

    Pr(Ben wins) = 1/6 + 1/6 * (5/6)^2 + 1/6 * (5/6)^4 + 1/6 * (5/6)^6 + … = 1/6 (1 + 5/6^2 + 5/6^4 + 5/6^6 +…)
    Pr(Jenny wins) = 1/6 * 5/6 + 1/6 * (5/6)^3 + 1/6 * (5/6)^5 + 1/6 * (5/6)^7 = 1/6 (5/6 + 5/6^3 + 5/6^5 + 5/6^7 + …)

    You notice something interesting here, which is that Ben has a higher chance of winning since (1 + 5/6^2 + 5/6^4 + 5/6^6 +…) > (5/6 + 5/6^3 + 5/6^5 + 5/6^7 + …) - each of the terms of the first sum is larger than the second sum.

    Both of these are geometric series - but are a bit complicated because they only involve every second power, so they're actually quite hard to compute. The best way to do them is to realise that a^(2b) = (a^2)^b.

    Therefore, (1 + 5/6^2 + 5/6^4 + 5/6^6 +…) = (1 + (25/36) ^ 1 + (25/36) ^ 2 + (25/36) ^ 3 + …) = 1 / (1 - 25/36) = 36 / 11 (infinite geometric series formula)
    And then, (5/6 + 5/6 ^ 3 + 5/6 ^ 5 + 5/6 ^ 7) = 5/6 (1 + 5/6 ^2 + 5/6 ^ 4 + 5/6 ^ 6 + …) = 5/6 * 36/11 (we just figured this out above).

    Therefore, after all that, what do you get?

    Pr(Ben wins) = 1/6 * 36 / 11 = 6 /11
    Pr(Jenny wins) = 1/6 * 5 / 6 * 36 / 11 = 5 /11

    And of course, they all sum to 11 / 11 = 1 - yes sir!!

    FWIW, this was a fun challenge, I loved it, if I was still teaching at a university, this could have been an exam question for my first year probability course - I'll keep it in the book ;)

    Have fun, and enjoy!

    • srr on 28/12/2022 - 00:58
      +2

      Let’s have a crack on how to get to 1/6:

      Ben and Jenny each roll a dice but keep result in secret. Then they throw a coin - if tails Jenny opens first, if heads Ben. The first to uncover 5 wins.

      • p1 ama on 28/12/2022 - 01:13
        +2

        Ben and Jenny each roll a dice but keep result in secret. Then they throw a coin - if tails Jenny opens first, if heads Ben. The first to uncover 5 wins.

        This still won't get you to 1/6 each though - I think this will get you to 11 / 36 * 1 / 2 = 11 / 72 each (which is juuuust ever so slightly below 1/6 each). This is intuitive, but let me try and see if this is actually right. (Intuition comes from the probability of a draw being the same).

        So there are now 8 possible cases:

        Cases where Ben wins:
        Pr(Ben 5, Jenny 5, H) = 1 / 6 * 1 / 6 * 1 / 2 = 1 / 72
        Pr(Ben 5, Jenny not 5, H) = 1 / 6 * 5 / 6 * 1 / 2 = 5 / 72
        Pr(Ben 5, Jenny not 5, T) = 1 / 6 * 5 / 6 * 1 / 2 = 5 / 72
        Total: 11/72 (just like I intuitively thought!!!)

        Cases where Jenny wins are just the reverse, so also 11/72

        There are then two draw cases:
        Pr(Ben not 5, Jenny not 5, H) = 5 / 6 * 5 / 6 * 1 / 2 = 25 / 72
        Pr(Ben not 5, Jenny not 5, T) = 5 / 6 * 5 / 6 * 1 / 2 = 25 / 72
        Total: 50 / 72 (or 25/36, same as in the original case).

        So therefore, the double check: 11/72 + 11/72 + 50/72 = 72/72 (voila)

        • srr on 28/12/2022 - 07:22
          +1

          Yes you’re right, they just split the probability of winning in case of rolling (5,5) between them. So they will have equal chances but not 1/6.

          They have to “win” both to get to 1/6, I.e. make the game completely independent. Removing “win” will help children understand that they both can have a positive outcome.

          Ben and Jenny roll a dice each once. They get to take a cookie from the cookie jar if they roll a 5. What is the probability of eating a cookie for them?

    • Kail on 28/12/2022 - 01:18
      +2

      What if winning wasn't considered mutually exclusive for the scenario where they are rolling at the same time?
      So if both roll 5, they both win?

      • p1 ama on 28/12/2022 - 01:27
        -1

        This is not possible within probability (i.e. both winning) - there is rigorous theory behind this which you'll learn in around a 3rd year undergrad probability course (or maybe a 1st year graduate), but without getting into measure theory, the reason is that it violates the sum of all probabilities being 1.

        In this case, what you would have is the probability of Ben / Jenny winning being 1 / 6 each, with the probability of a draw being 25 / 36, which adds up to more than 1.

        You can modify the parameters a little to get it to work, e.g. you can say that if they both roll a 5, we flip a coin to see who wins, but again, this will not give you 1/6, it'll give you 11/72 each (as per the above explanation: https://www.ozbargain.com.au/comment/13169028/redir)

        Think about a simpler example - we both flip a coin, "whoever flips H will win" (which is bad language, but let's run with it). Obviously we both have a 1/2 probability of flipping a H, but there's a 1/4 probability we both flip a T, so we cannot just say that "we both win" if we both flip H.

        • Kail on 28/12/2022 - 01:33
          +2

          But it's a valid game rule, the players can choose to play the game this way.
          Are you saying the game would be a bad example for a probability question?
          I feel like that's a summary of this entire thread tbh. The question the teacher was posing to the fifth graders wasn't really a good choice.

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