GCSE Maths Tree Diagrams Notes
Tree diagrams organise two-stage probability problems. Multiply along branches for 'and', and add the relevant branch results for 'or'.
Key concepts
- Branches
- Each branch shows an outcome and its probability. Probabilities on branches from one point add to 1.
- Multiply along
- To find the probability of a combined path (this AND that), multiply along the branches.
- Add between
- To combine separate paths (this OR that), add the path probabilities.
- Without replacement
- If an item is not replaced, the second-stage probabilities change (the total drops by one).
Key formulas
Combined path
P(A then B) = P(A) × P(B)
Either path
P(path 1 or path 2) = P(path 1) + P(path 2)
Worked example
A bag has 4 red and 6 blue counters. Two are taken without replacement. Find P(both red).
- 1First counter: P(red) = 4/10.
- 2Without replacement, 3 red remain of 9: P(red) = 3/9.
- 3Multiply along the branches: 4/10 × 3/9 = 12/90.
- 4Simplify: 2/15.
Answer: 2/15
Common mistakes
✗ Keeping the same probabilities for 'without replacement'.
✓ Reduce both the favourable count and the total by one for the second pick.
✗ Adding along a single path.
✓ Multiply along a path; only add when combining different paths.
✗ Branch probabilities that do not sum to 1.
✓ From each point, the branch probabilities must total 1.
Quick check
Try these, then reveal the answers.
Along one path, do you add or multiply?Reveal answer
Multiply.
P(head then head) with a fair coin?Reveal answer
1/4
After taking 1 red from 4 red / 6 blue (no replacement), P(red) next?Reveal answer
3/9 = 1/3
Exam tips
- Label every branch with its probability before calculating.
- For 'at least one', it is often quicker to do 1 − P(none).
- Simplify final fractions and check they are less than 1.
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Frequently asked questions
When do probabilities change on a tree diagram?+
When there is no replacement — the total and the relevant count both fall by one for the next stage.
How do I find 'at least one'?+
Work out the probability of none happening and subtract from 1.
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