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Notes for Teachers

Notes for Teachers. This activity is based upon Non-Transitive Dice, and is an excellent exploration into some seemingly complex probability.

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Notes for Teachers

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  1. Notes for Teachers • This activity is based upon Non-Transitive Dice, and is an excellent exploration into some seemingly complex probability. • The three dice version has been around for a while, but with different numbers on the dice. The version here is so it fits with the 5 dice version. (If you have a three dice set, the probabilities in each case are the same, it is just the numbers on the dice that need changing in the Tree Diagrams) • It is best done using the Non-Transitive Dice, which you can buy from http://mathsgear.co.uk/collections/dice/products/non-transitive-grime-dice • You could also make the dice as a starter activity, and a recap on nets (just use different coloured card, and remember to put the correct numbers on each die). • The slides talk the students through what they need to do, and I have put some comments on ideas for questions and practicalities in the notes box. • The Grime dice (5 dice set) were discovered by James Grime of the University of Cambridge, and his video description and article can be found at http://grime.s3-website-eu-west-1.amazonaws.com/ • This slideshow is an attempt at a teacher friendly, usable in the classroom, way of presenting this information. • The spreadsheet calculates all the probabilities and allows users to change the values on the dice. • There is another great way to introduce Non-Transitive dice at http://nrich.maths.org/7541 • For more interactive resources, visit my website at http://www.interactive-maths.com/

  2. Dice Games In your pairs, you are going to play a game. You each have a coloured die, and you are going to both throw your die. The player with the highest score wins that round. Play 10 rounds. Who is winning overall? Play a further 90 rounds (100 in total). Is the game fair?

  3. What did we discover? How did RED and BLUE compete? We saw that RED beats BLUE. How did BLUE and GREEN compete? We saw that BLUE beats GREEN. What do we expect in the REDvsGREEN games? We expect that since RED beats BLUE and BLUE beats GREEN, then RED will beat GREEN. This is called a Transitive Property – the win is transferred through the blue! Numbers are transitive: if 5 > 3 and 3 > 1, then 5 > 1!

  4. What actually happened in the RED and GREEN games? We see that GREEN beats RED. Non-Transitive Dice BEATS BEATS BEATS

  5. Why is it that this happens? Let’s take a look at the probabilities! First we need to know what numbers are on each die. 0, 5, 5, 5, 5, 5 2, 2, 2, 7, 7, 7 4, 4, 4, 4, 4, 9 Now we can use our knowledge of probabilities to calculate the probability in each battle. We shall use a tree diagram to consider the multiple outcomes.

  6. 0, 5, 5, 5, 5, 5 2, 2, 2, 7, 7, 7 4, 4, 4, 4, 4, 9 REDvsBLUE RED BLUE 2 4 So RED wins over BLUE with probability 7 2 9 7

  7. 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 Use the values on the three die to make two further Tree Diagrams to show that the Dice are indeed Non-Transitive.

  8. 0, 5, 5, 5, 5, 5 2, 2, 2, 7, 7, 7 4, 4, 4, 4, 4, 9 BLUEvsGREEN GREEN BLUE 0 2 So BLUE wins over GREEN with probability 5 0 7 5

  9. 0, 5, 5, 5, 5, 5 2, 2, 2, 7, 7, 7 4, 4, 4, 4, 4, 9 GREENvsRED RED GREEN 4 0 So GREEN wins over RED with probability 9 4 5 9

  10. Pair up with somebody with the same colour die as you. Now make a group of 4 by joining another pair (there should be two dice of two different colours in your group). We are going to play the game again, but taking the total of the same coloured dice. Play 100 rounds as before, and keep track of how many rounds each colour wins.

  11. What did we discover this time? How did RED and BLUE compete? We saw that BLUE beats RED. How did BLUE and GREEN compete? We saw that GREEN beats BLUE. How did GREEN and RED compete? We saw that RED beats GREEN. This is the opposite to what happened with only one die of each colour!!!

  12. With two dice, the rules are a little bit different! BEATS BEATS BEATS Let’s have a look at the probabilities again!

  13. 0, 5, 5, 5, 5, 5 2, 2, 2, 7, 7, 7 4, 4, 4, 4, 4, 9 REDvsBLUE (two dice) 4 9 8 14 So BLUE wins over RED with probability 4 13 9 14 4 18 9 14

  14. 0, 5, 5, 5, 5, 5 2, 2, 2, 7, 7, 7 4, 4, 4, 4, 4, 9 BLUEvsGREEN (two dice) 0 5 4 10 So GREEN wins over BLUE with probability 0 9 5 10 0 14 5 10

  15. 0, 5, 5, 5, 5, 5 2, 2, 2, 7, 7, 7 4, 4, 4, 4, 4, 9 GREENvsRED (two dice) 8 13 0 18 So RED wins over GREEN with probability 8 5 13 18 8 10 13 18

  16. SUMMARY One Die BEATS BEATS Two Dice 4, 4, 4, 4, 4, 9 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 Remember the word lengths get bigger: RED (3) -> BLUE (4) -> GREEN (5) BEATS BEATS How to Use this Game Place the three dice out, and get a friend to play. Ask them to choose a die to use, and you then pick the one which will beat it. Role the dice 20 times, and you should win. Once they think they have worked it out, agree to take the die first. When they pick a die, if you are to win, leave it be, but if you are to lose say that you want to “double the stakes” with a second die each. This reverts the order! BEATS BEATS

  17. 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 3, 3, 3, 3, 8, 8 1, 1, 6, 6, 6, 6 4, 4, 4, 4, 4, 9 This is a set of 5 Non-Transitive Dice What do you notice about the dice? The numbers 0-9 appear on exactly 1 die. The 3 dice set is included within the 5 dice set. This set of dice are called Grime Dice, after their discoverer, James Grime at the University of Cambridge

  18. As with the 3 dice set, we can work out the probabilities in each pairing. How many different ways could we pair up the different coloured dice? 4 RED with each of BLUE, OLIVE, YELLOW and MAGENTA 3 BLUE with each of OLIVE, YELLOW and MAGENTA 2 OLIVE with each of YELLOW and MAGENTA 1 YELLOW with MAGENTA So there are 10 possible pairings! We use OLIVE and MAGENTA instead of green and purple for a good reason we shall see!!!. We need to look at all of them!

  19. Each pair has been given a colour pair to look at. Use a tree diagram to calculate the probabilities involved, and which colour will win. We already know three: RED > BLUE with probability BLUE > OLIVE with probability OLIVE > RED with probability

  20. And now for the full list of all the probabilities……… Colour names get longer Colour names are alphabetical There are 2 chains that work for the 5 dice What do you notice? How do the names relate to the chains? How do the probabilities compare?

  21. BEATS BEATS BEATS BEATS BEATS BEATS BEATS BEATS BEATS BEATS

  22. Notice that we can make several sets of 3 Non-Transitive dice by following paths on this graph. Each of these 5 subsets of dice will produce a valid set of 3 Non-Transitive Dice. They are obtained by taking 3 consecutive dice in the Word Length list.

  23. We can also make sets of 4 Non-Transitive Dice! Each of these 5 subsets of dice will produce a valid set of 4 Non-Transitive Dice. They are obtained by taking 4 consecutive dice in the Alphabetical list.

  24. Combine two pairs to make a group of 4 people, with 10 dice! In your group, investigate what happens in the different combinations available when each pair has 2 dice (of the same colour). You can use a mixture of experimental probabilities and theoretical probabilities.

  25. And now for the full list of all the probabilities……… Colour names get longer Colour names are alphabetical The Word Length Chain is reversed as expected. What do you notice? The Alphabetical Chain is in the same order How do the probabilities compare?

  26. This line is 50:50 either way

  27. SUMMARY 2, 2, 2, 7, 7, 7 0, 5, 5, 5, 5, 5 3, 3, 3, 3, 8, 8 1, 1, 6, 6, 6, 6 4, 4, 4, 4, 4, 9 One Die Two Dice Word Length Alphabetical How to Use this Game Place the three dice out, and get a friend to play. Ask them to choose a die to use, and you then pick the one which will beat it. Role the dice 20 times, and you should win. Once they think they have worked it out, agree to take the die first. When they pick a die, if you are to win, leave it be, but if you are to lose say that you want to “double the stakes” with a second die each. This reverts the order!

  28. In your groups you are going to create a poster on Non-Transitive Dice. Colour Title Background Info Some of the Maths Presentation Challenges Succinct Layout

  29. A Special Game We can now use the set of 10 dice to play two players at once, and improve our chance of beating both of them Invite two opponents to pick a die each, but do NOT say whether you are playing with one die or two. If you opponents pick two dice that are next to each other on the alphabetical list (not next to each other around the circle), then play the one die game, and use the diagram to choose the die that will beat both most of the time. If you opponents pick two dice that are next to each other on the word length list(next to each other around the circle), then play the two dice game, and use the diagram to choose the die that will beat both most of the time.

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