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 Answer 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.