# MARS: Analysing games of chance

The ‘Analysing games of chance’ resource is a concept development lesson from the ‘Mathematics Assessment Resource Service’. The resource is designed to confront and overcome common probability misconceptions.

The main activity in the resource is a game of ‘Horse Race’. Students are given the following board:

The game has the following rules:

- Place counters on the starting squares labelled 2 to 12.
- Roll two dice and add the scores. The horse with that number moves one square forward.
- Keep rolling the dice, the horse that is first past the finishing line wins.
- Stop rolling the dice when the first horse wins and record the position of all the horses.

Students have to predict which horse they think will win. They then play the game several times to gain some empirical evidence. The results from the class can be pooled to provide stronger evidence.

A distinctive feature of the resource is the extensive guidance that it provides for the teacher. Suggested questions to ask at this stage include:

- If you were able to play the game again, which horse is most likely to win?
- Which horse is most likely to come second?
- Which horses are least likely to win?

Students then look at the theoretical probability behind the race. Here is a probability space diagram to represent the situation:

The number 7 occurs with greater frequency than the other numbers, so horse 7 is the one most likely to win the race.

There are two examples of student work provided with the resource. These can be used for peer assessment.

In the first example the student has attempted to list all of the combinations of the two dice. Unfortunately, they have not distinguished between 4 + 1 and 1 + 4, so whilst their conclusion is correct, there are mistakes in their reasoning.

In the second piece of work, the student has drawn a correct probability space diagram, but reasoned incorrectly that horse 7 will win one-sixth of the time.

If extending the work to a second lesson, there is an opportunity for students to apply what they have learned to three different versions of the horse race game. Different rules for moving forward include:

- Find the difference between the scores on the dice.
- Use the highest number shown on the two dice.
- Multiply the numbers shown on the dice and test to see if it is a multiple of the horse number.

In addition to the horse race activity, the resource includes a formative assessment task ‘Spinner Bingo’ which is completed before the main part of the lesson. This gives an opportunity to discover misconceptions that students have.

- ANALYZING GAMES OF CHANCE resource
- The Mathematics Assessment Resource Service (MARS) is a collaboration between the University of California at Berkeley and the Shell Centre team at the University of Nottingham, with support from the Bill and Melinda Gates Foundation. The team is known around the world for its innovative work in maths education. Previous projects that members of the team have worked on include the DfE Standards Units and the Bowland Maths resources.