```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.
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’.
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?
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
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
Is it possible to put values on the dice so that the
different totals obtained are all equally likely?
Die A
+
?
Die B
?
?
?
?
?
?
?
?
?
?
?
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
Could you make two different totals?
Dice A
+
?
Dice B
?
?
?
?
?
?
?
?
?
?
?
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.
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?
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.
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.
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
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
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
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
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.
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
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
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.
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?
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
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.
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
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.
» Students should have the opportunity to discuss this
with a partner or in a small group
» Students should sketch or calculate (as appropriate)
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?
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.