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All About Dice Dice have been around for at least 5000 years and are used in many games. Knucklebones of animals, which are approximately tetrahedral in shape were used – and still are in some countries. All About Dice Six sided dice have become common place, but it is possible to make dice with different numbers of sides. Platonic solids are often used as the regularity of their shape makes them ‘fair’. All About Dice When rolling 2 six-sided dice, how many different possible totals are there? Are they all equally likely? Rolling a double is often the way to start a game, or to get out of some difficulty, how likely is it that you will roll a double? Rolling a ‘double six’ is even more challenging, how likely is it? All About Dice Die B One way to obtain all the possible outcomes is through using a two-way table. + 1 2 Die A 3 4 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 5 6 All About Dice Die B It is obvious from the table that the chance of obtaining a total of 12 is far less than the chance of obtaining a total of 7. + 1 2 Die A 3 4 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 5 6 All About Dice Is it possible to put values on the dice so that the different totals obtained are all equally likely? Die A + ? Die B ? ? ? ? ? ? ? ? ? ? ? All About Dice This is one way, but it’s probably not very helpful for use in a game for everyone to always roll the same! Die B Die A + 1 1 1 1 1 1 2 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 3 3 3 3 All About Dice Could you make two different totals? Dice A + ? Dice B ? ? ? ? ? ? ? ? ? ? ? All About Dice This is a solution… Die B Die A + 1 1 1 2 2 2 2 3 3 3 4 4 4 2 3 3 3 4 4 4 2 3 3 3 4 4 4 2 3 3 3 4 4 4 2 3 3 3 4 4 4 2 3 3 3 4 4 4 However, it’s probably not that useful for a game, and die B is redundant because whether you get the higher or lower total is determined by die A. All About Dice There are 36 different outcomes and so far we have seen: • 1 total in 36 different ways • 2 totals, each in 18 different ways Thinking about the factors of 36, the number of different outcomes could potentially be: 1 2 3 4 6 9 12 18 36 Using any numbers, could you devise 2 dice so that there are 36 different totals? Can they be 1 to 36? All About Dice Here’s one solution: Die B Die A + 1 2 3 4 5 6 0 1 2 3 4 5 6 6 7 8 9 10 11 12 12 13 14 15 16 17 18 18 19 20 21 22 23 24 24 25 26 27 28 29 30 30 31 32 33 34 35 36 Again, it’s probably not that useful for a game. All About Dice Which of the following could dice be created to give: • 3 totals, each in 12 different ways? • 4 totals, each in 9 different ways? • 6 totals, each in 6 different ways? • 9 totals, each in 4 different ways? • 12 totals, each in 3 different ways? • 18 totals, each in 2 different ways? A restriction is that neither die can have the same number on all 6 faces. All About Dice Here are some solutions, but there are many others: Die A + 1 2 3 4 5 6 + 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 6 7 8 9 10 11 12 0 1 2 3 4 5 6 6 7 8 9 10 11 12 6 7 8 9 10 11 12 12 13 14 15 16 17 18 6 7 8 9 10 11 12 12 13 14 15 16 17 18 6 7 8 9 10 11 12 Die B Die B Die A All About Dice Here are some solutions, but there are many others: Die A + 1 1 2 2 3 3 + 1 1 2 2 3 3 0 1 1 2 2 3 3 0 1 1 2 2 3 3 0 1 1 2 2 3 3 0 1 1 2 2 3 3 3 4 4 5 5 6 6 0 1 1 2 2 3 3 3 4 4 5 5 6 6 3 4 4 5 5 6 6 6 7 7 8 8 9 9 3 4 4 5 5 6 6 6 7 7 8 8 9 9 3 4 4 5 5 6 6 Die B Die B Die A All About Dice Here are some solutions, but there are many others: Die B Die A + 1 1 1 2 2 2 0 1 1 1 2 2 2 0 1 1 1 2 2 2 0 1 1 1 2 2 2 2 3 3 3 4 4 4 2 3 3 3 4 4 4 2 3 3 3 4 4 4 All About Dice In a very simple dice game, you and your opponent choose a die each and then roll it. The one who rolls the highest number wins. It couldn’t be simpler! On the next slide are the nets of the 3 dice. Which one would you choose and why? 6 9 4 2 9 4 1 2 8 6 8 7 1 5 3 7 5 3 All About Dice You might like to make the dice and test them out to see which one wins most often. Checking it mathematically, we could again use twoway tables. Complete the tables on the following slides, recording which dice would win each time. All About Dice Blue dice Yellow dice vs 2 2 4 4 9 9 1 Y Y Y Y Y Y 1 Y Y 6 B B 6 B 8 B 8 Yellow wins ___ times Blue wins ___ times All About Dice Red dice Blue dice 1 1 6 6 8 vs 8 3 2 3 2 5 5 7 7 Blue wins ___ times Red wins ___ times Yellow dice Red dice vs 3 3 5 5 7 7 4 4 9 9 Red wins ___ times Yellow wins___ times All About Dice This might seem quite unusual: • Yellow is better than blue • Blue is better than red • Red is better than yellow This is a special set of dice and such sets are called ‘non-transitive dice’. It’s a bit similar to the ‘Rock, paper, scissors’ game where there is no best item overall, it depends on what the other person has. All About Dice In a variation of the game, there are 3 players who each have one of the dice. They roll the dice at the same time, the highest number wins. Which die would you choose? You might like to make the dice to test them. Can you think of a way to find all the possible outcomes? All About Dice One approach is to construct a tree diagram, as shown on the following slide, another is to list all the outcomes systematically in a table. Each number appears twice on each dice, but has only been listed once as they are all equally likely. (If we listed them twice, we’d simply repeat the tree or the table; would this make a difference to the final probabilities?) 1 2 6 8 1 roll 4 6 8 1 9 6 8 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7 3 5 7 Check along each of the branches to find out which colour wins each time. Yellow Blue 2 1 2 1 2 1 2 6 2 6 2 6 2 8 2 8 2 8 Red 3 5 7 3 5 7 3 5 7 4 4 4 4 4 4 4 4 4 1 1 1 6 6 6 8 8 8 3 5 7 3 5 7 3 5 7 9 9 9 9 9 9 9 9 9 1 1 1 6 6 6 8 8 8 3 5 7 3 5 7 3 5 7 All About Dice The result for this may also be surprising. Yellow and Blue are equally as good as each other, winning 10 times out of 27 and Red is slightly worse, winning 7 times out of 27. All About Dice There are other sets of numbers that create nontransitive dice. Another set of 3 non-transitive dice are given on the next slide and then a set of 4 non-transitive dice are on the following slide. These are known as ‘Efron’s Dice’ after Bradley Efron who devised them. They are given in order, so that each die is beaten by the previous one. What is the probability of winning each time? If all 3 (or all 4) dice are used, is there a ‘best’ or worst’ die? 5 6 7 2 6 7 1 2 9 5 9 8 1 4 3 8 4 3 3 4 0 4 4 0 3 3 4 3 3 5 6 3 2 2 6 2 1 2 1 5 5 1 Teacher notes: All about dice This activity looks at six sided dice and students consider variations of numberings including sets of ‘non-transitive’ dice. Many of the activities are problem solving in nature. It is assumed that students are familiar with the outcomes when rolling two dice, but the activity begins with a few questions to consider to recap on this before moving on. Teacher notes: All about Dice » Students should have the opportunity to discuss this with a partner or in a small group » Students should sketch or calculate (as appropriate) Teacher notes: All about dice Slide 4: students should discuss. This is a short opportunity to recap on previous learning. Answers: not all equally likely. Chance of a double is 6/36 (or 1/6). Chance of a double six is 1/36. Slide 13: Different groups of students could be set different ones to tackle. This problem solving activity is trying to help students really understand the structure of what happens to the totals. Questions: • Is it possible to make all the totals consecutive? • Is it possible to make all the totals consecutive even numbers? Teacher notes: All about dice Slide 18: students should discuss and justify their responses. This should prove interesting as there is no ‘best’ die. The ‘right’ answer is that you should choose your die based on what your opponent has chosen – similar to Rock, paper, scissors – which students may be familiar with. (As an aside, The Big Bang Theory explains a development of this game: Rock, Paper, Scissors, Lizard, Spock.) Each dice beats the subsequent one with a probability of 5/9 Slide 28: the set of 3 dice are very similar to the original set of 3 in that they also have a probability of beating the subsequent on with a probability of 5/9, but this time there is one die which is best if all 3 are thrown together – the blue die has an 11/27 chance of winning whilst the other two each have an 8/27 chance of winning. Teacher notes: All about dice Slide 28: (cont) with the 4 dice, each one beats the following one with a probability of 2/3 of winning. If all 4 dice are rolled, C will win 4/9 of the time, D will win 3/9 of the time and A will win 2/9 of the time. B will never win. Acknowledgements Dice pictures from http://www.laputanlogic.com/articles/2004/12/0110001-6358.html Pictures and info http://en.wikipedia.org/wiki/Dice#mediaviewer/File:Knuck_dice_Steatite _37x27x21_mm.JPG