Tree diagrams and combined events

Tree diagrams show every outcome of two or more events. Multiply along the branches for “and”, and add the end results for “or”.

Key facts

  • Probabilities on each pair of branches add to 1.
  • Multiply along branches: P(A and B).
  • Add the outcomes you want: P(this or that).
  • Without replacement, the second set of probabilities changes.

Worked examples

Example 1

A bag has 3 red and 2 blue balls. Two are taken without replacement. Find P(one of each colour).

  1. Red then blue: 3/5 × 2/4 = 6/20.
  2. Blue then red: 2/5 × 3/4 = 6/20.
  3. Add: 12/20 = 3/5.

Answer: 3/5

Example 2

P(bus late) = 0.2 each day, independently. Find P(late on at least one of two days).

  1. P(never late) = 0.8 × 0.8 = 0.64.
  2. 1 − 0.64 = 0.36

Answer: 0.36