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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.

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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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