Key Idea: Topic 4.6 builds on the probability rules from 4.5 by giving you two powerful visual tools: Venn diagrams (for showing how sets overlap and how probabilities relate) and tree diagrams (for sequential events where the outcome of the first affects the second). Both tools organise information — the key is knowing when to use each one.
✅ Venn diagrams
✅ Tree diagrams for conditional probability
Example: Venn diagram: In a class of 30: 18 study French (F), 12 study Spanish (S), 7 study both. n(F∩S) = 7. n(F only) = 18−7 = 11. n(S only) = 12−7 = 5. n(neither) = 30−(11+7+5) = 7. P(F|S) = P(F∩S)/P(S) = (7/30)/(12/30) = 7/12 Tree diagram without replacement: 4 red, 3 blue balls. Draw two without replacement. P(red then blue) = (4/7) × (3/6) = 12/42 = 2/7
When filling a Venn diagram: always start with the intersection (A∩B), then fill in 'only A' and 'only B', then 'neither'. This order prevents double-counting. All probabilities in a tree diagram (summing across all final branches) must add to 1. Use this as a check.
Paper 1: You may be given a partially completed Venn diagram and asked to find a probability. Read n(A∩B) from the overlap region and use the conditional probability formula. Paper 2: Multi-stage probability problems almost always benefit from a tree diagram. Show all branches with their probabilities, then highlight the relevant paths.