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SQA National 5 Mathematics

Tree diagrams

Using branches to calculate combined probabilities.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Probabilities on branches from the same point should add to 1.
  • Multiply along branches.
  • Add separate routes for an OR outcome.
National 5 Mathematics lesson

Explanation

Tree diagrams organise multi-stage probability questions.

For each complete route, multiply the probabilities along the branches.

If more than one route gives the required outcome, add the route probabilities.

Visual support

Multiply along branches and add routes

startR 3/5B 2/5R 2/4B 2/4R 3/4B 1/4P(R then B) = 3/5 x 2/4

Key formulae and rules

  • P(A and B) = P(A) x P(B after A)
  • P(A or B) = add separate successful routes
  • P(not A) = 1 − P(A)

Watch out

Adding along a branch instead of multiplying.

Check

Compare your answer with the size you expected from the question.

Exam tip

Write the route letters, such as RW and WR, before multiplying. This helps avoid missed cases.

Calculator tip

Use brackets for fractions, powers and square roots, then round only at the final line.

Worked examples

Worked example 1

With replacement

A bag has 3 red and 2 blue counters. One counter is chosen and replaced, then another is chosen. Find P(red then blue).

  1. P(red) = 35
  2. Replacement means P(blue) = 25 again
  3. Multiply: 35 × 25 = 625

Answer: P(red then blue) = 625

Worked example 2

Without replacement

A bag has 4 green and 1 yellow counters. Two counters are chosen without replacement. Find P(two green).

  1. P(first green) = 45
  2. After one green is removed, P(second green) = 34
  3. Multiply: 45 × 34 = 35

So: P(two green) = 35

Worked example 3

At least one success

A spinner has P(win) = 14. It is spun twice. Find P(at least one win).

  1. Use complement: P(no wins) = 34 × 34 = 916.
  2. P(at least one win) = 1 − 916

Answer: P(at least one win) = 716

Watch out

  • Adding along a branch instead of multiplying.
  • Forgetting probabilities change without replacement.
  • Missing one route when the wording says at least one.