EST. 2024 · LONDON·MMXXVI SPECIFICATION
AQA·Edexcel·OCR|Foundation + Higher
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Statistics & Probability · Foundation & Higher

Tree diagrams

Tree diagrams show all possible outcomes of two or more events and their probabilities. You multiply along branches and add between branches.

Revising tree diagrams the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Tree diagrams GCSE” on the site for the complete guide.

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Key facts to remember

  • 1Multiply along branches to get the probability of combined outcomes.
  • 2Add probabilities for mutually exclusive outcomes.
  • 3All probabilities on branches from one point must add up to 1.
  • 4For "without replacement", the probabilities on the second set of branches change.
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Worked examples

Example 1

A bag has 4 red and 6 blue balls. A ball is drawn, replaced, then another is drawn. Find P(both red).

Working

  1. P(red) = 4/10 = 2/5 on both occasions (with replacement)
  2. P(both red) = 2/5 × 2/5 = 4/25
AnswerP(both red) = 4/25
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Common mistakes

✗Adding instead of multiplying along branches.
✗Forgetting to update probabilities for "without replacement" questions.
✗Not checking that branches at each node sum to 1.
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Exam tips

✓Draw the tree clearly with all outcomes and probabilities labelled on each branch.
✓Circle or highlight the branches you need before multiplying.

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