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Article by Jenny Gage# Tree Diagrams, 2-way Tables and Venn Diagrams

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

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

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

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Age 11 to 18

Published 2013

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

- Collect data, which is represented on a tree diagram and 2-way table, using whole numbers (natural frequencies).
- Use the natural frequencies to derive proportions for each outcome.
- Consider what the proportions will settle down to, as more data accumulates.
- Compare the experimental results and proportions with the expected results and proportions (which are the limits the data should approach as more is collected).
- 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.

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

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

- 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)