This article is part of our collection Great Expectations: Probability through Problems.

Our approach to diagrammatic representations for probability is for students to:

1. Collect data, which is represented on a tree diagram and 2-way table, using whole numbers (natural frequencies).
2. Use the natural frequencies to derive proportions for each outcome.
3. Consider what the proportions will settle down to, as more data accumulates.
4. Compare the experimental results and proportions with the expected results and proportions (which are the limits the data should approach as more is collected).
5. Normalise (so find the equivalent fraction of 1) the expected proportions to give the probabilities of each event, and hence the probability of each outcome.

We do not make use of Venn diagrams in this process, but since they are part of many curricula for 11-16 (and 16+) students, we include them in this article.

There is a progression for each representation which students need to go through:

1. Natural frequencies.
2. Proportions derived from natural frequencies.
3. Probabilities.
4. Generalisation of each of these.
5. Reverse tree diagrams.
6. Hence see where the 'rules' of probability come from - specifically, the multiplication rule and Bayes' Theorem.

Examples of the general process and the process for each of the main Probability from Problems resources are linked:

• Representations in general
• Which Team Will Win? - problem, teachers' notes, representations and answers (11-13 year-olds)
• The Dog Ate My Homework - problem, teachers' notes, representations and answers (12-14 year-olds)
• Who Is Cheating? - problem, teachers' notes, representations and answers (13-15 year-olds)
• Prize Giving - problem, teachers' notes, representations and answers (14-16 year-olds)