This activity is based upon NonTransitive 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 se ID: 264435
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Slide1
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/
Slide2
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?Slide3
What did we discover?
We saw that
RED
beats BLUE.
How did RED and BLUE compete?
We saw that BLUE beats GREEN.
How did BLUE and GREEN compete?
What do we expect in the RED vs GREEN 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!Slide4
We see that
GREEN
beats RED.
What actually happened in the RED and GREEN games?
BEATS
BEATS
BEATS
Non-Transitive DiceSlide5
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.
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 7
0, 5, 5, 5, 5, 5
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.Slide6
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 7
0, 5, 5, 5, 5, 5
RED vs
BLUE
RED
4
9
BLUE
2
7
2
7
So
RED
wins over
BLUE
with probability
Slide7
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.Slide8
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 7
0, 5, 5, 5, 5, 5
BLUE vs
GREEN
BLUE
2
7
GREEN
0
5
0
5
So
BLUE
wins over
GREEN
with probability
Slide9
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 7
0, 5, 5, 5, 5, 5
GREEN vs
RED
GREEN
0
5
RED
4
9
4
9
So
GREEN
wins over
RED
with probability
Slide10
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.Slide11
What did we discover this time?
We saw that
BLUE
beats RED.
How did RED and BLUE compete?
We saw that GREEN beats BLUE.
How did BLUE and GREEN compete?
We saw that RED beats GREEN.
How did GREEN and RED compete?
This is the opposite to what happened with only one die of each colour!!!Slide12
With two dice, the rules are a little bit different!
BEATS
BEATS
BEATS
Let’s have a look at the probabilities again!Slide13
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 7
0, 5, 5, 5, 5, 5
RED vs
BLUE(two dice)
8
18
4
14
4
14
13
9
9
4
14
9
So
BLUE
wins over
RED
with probability
Slide14
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 7
0, 5, 5, 5, 5, 5
BLUE vs
GREEN(two dice)
4
14
0
10
0
10
9
5
5
0
10
5
So
GREEN
wins over
BLUE
with probability
Slide15
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 7
0, 5, 5, 5, 5, 5
GREEN vs
RED(two dice)
0
10
8
18
8
18
5
13
13
8
18
13
So
RED
wins over
GREEN
with probability
Slide16
SUMMARY
BEATS
BEATS
BEATS
One Die
BEATS
BEATS
BEATS
Two Dice
Remember the word lengths get bigger:
RED (3) -> BLUE (4) -> GREEN (5)
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!
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 7
0, 5, 5, 5, 5, 5Slide17
This is a set of 5 Non-Transitive Dice
What do you notice about the dice?
The 3 dice set is included within the 5 dice set.
The numbers 0-9 appear on exactly 1 die.
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 70, 5, 5, 5, 5, 5
3, 3, 3, 3, 8, 8
1, 1, 6, 6, 6, 6This set of dice are called Grime Dice, after their discoverer, James Grime at the University of CambridgeSlide18
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?
RED with each of
BLUE, OLIVE, YELLOW and MAGENTA
BLUE with each of OLIVE,
YELLOW and MAGENTA
OLIVE with each of YELLOW and MAGENTA
YELLOW with MAGENTA
4
3
2
1
We use
OLIVE
and
MAGENTA
instead of green and purple for a good reason we shall see!!!.
So there are 10 possible pairings!
We need to look at all of them!Slide19
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 Slide20
And now for the full list of all the probabilities………
What do you notice?
There are 2 chains that work for the 5 dice
How do the names relate to the chains?
Colour names get longer
Colour names are alphabetical
How do the probabilities compare?Slide21
BEATS
BEATS
BEATS
BEATS
BEATS
BEATS
BEATS
BEATS
BEATS
BEATSSlide22
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.Slide23
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.Slide24
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.Slide25
And now for the full list of all the probabilities………
What do you notice?
The Word Length Chain is reversed as expected.
The Alphabetical Chain is in the same order
Colour names get longer
Colour names are alphabetical
How do the probabilities compare?Slide26
This line is 50:50 either waySlide27
SUMMARY
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!
One Die
Two Dice
4, 4, 4, 4, 4, 9
2, 2, 2, 7, 7, 7
0, 5, 5, 5, 5, 5
3, 3, 3, 3, 8, 8
1, 1, 6, 6, 6, 6
Word Length
AlphabeticalSlide28
In your groups you are going to create a poster on Non-Transitive Dice.
Colour
Title
Background Info
Some of the Maths
Challenges
PresentationSuccinct
LayoutSlide29
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.