# Dice

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```					Problem Solving – Unit 1
Probability

Dice
Three dice are numbered 1 to 6
Two of them are red and one is blue.
All three dice are rolled.

What is the probability that the total on the two red dice will be equal to the
score on the blue dice?
Notes for Teachers

This problem was originally written by Leeds University Assessment and Evaluation unit as a teaching
resource for the pilot GCSE in Additional mathematics as part of the QCDA funded Curriculum Pathways
Project. We are grateful to QCDA for giving their permission to reproduce these resources to support teachers
of GCSE mathematics.

Problem solving classification                      Set out cases

Content area classification                         Probability

Tier                                                Higher

5
72

About the question            This problem is about the ability to handle cases carefully –
dividing up the sample space in an appropriate way to enable the
question to be answered effectively. It appears deceptively
simple but without bringing additional structure to the problem it
is very difficult to tackle efficiently.
Problem-solving               The students must ask themselves how this problem can be
approaches                    broken down into parts that are amenable to the sorts of
techniques that they are familiar with in more straightforward
examples of probability. Will tree diagrams help? Is a 3-way
sample space do-able? How many actual equally likely outcomes
are there? Amongst these, can the desired outcomes simply be
counted?
Challenges / issues           The difficulty lies in the added layer of complexity beyond that
reached in more standard questions, in that there are three dice,
rather than just two. How the techniques that work for two dice
can be extended to this more difficult situation is the key to
finding the solution.
Finding the answer            1. This item can be tackled in a very probabilistic way using, for
example, a tree diagram starting with the single blue die and
this will involve lots of fractional calculations. Such a method,
carefully constructed with irrelevant parts of the tree diagram
properly omitted, can bring success, though does require
great attention to detail to cover all the cases correctly.
2. An alternative, more counting-based approach would be to
consider the two red dice as a whole and then to match the
scores on this pair to the single score on the blue die. This
method has the effect of bringing the complexity of the
problem down a level by making it into a problem requiring
the calculation of scores on a pair of dice matching the scores
on a single die. There is still work to be down in determining
the correct sample space for the pair of red dice and then
accounting for the matching on the blue but this is within
student’s experience of more standard problems.
Follow up   A natural extension that might encourage similar problem-solving
strategies could include matching scores on two pairs of six-
sided dice, or using three die, but not all six-sided.

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