Branching out
A tree diagram organizes the outcomes of an experiment with more than one stage. Each branch shows one possible outcome, and each complete path from start to end is one outcome in the sample space.
Example: two coin flips
Stage 1: H or T. From each, stage 2: H or T.
- H → H (HH)
- H → T (HT)
- T → H (TH)
- T → T (TT)
4 outcomes. Each path has probability ½ × ½ = ¼.
Probabilities on branches
Write the probability on each branch. Then:
- Along a path: multiply.
- Combining paths (for "or"): add.
Example: A spinner is ⅓ red and ⅔ blue. Spin twice.
| Path | Probability |
|---|---|
| R, R | ⅓ × ⅓ = 1/9 |
| R, B | ⅓ × ⅔ = 2/9 |
| B, R | ⅔ × ⅓ = 2/9 |
| B, B | ⅔ × ⅔ = 4/9 |
Check: all paths add to 1 ✓.
P(one of each colour) = 2/9 + 2/9 = 4/9
Dependent events
Branches at the second stage can have different probabilities.
Example: A bag has 3 green and 2 yellow marbles. Draw two without replacement.
- G (3/5) → G (2/4) or Y (2/4)
- Y (2/5) → G (3/4) or Y (1/4)
| Path | Probability |
|---|---|
| G, G | 3/5 × 2/4 = 6/20 |
| G, Y | 3/5 × 2/4 = 6/20 |
| Y, G | 2/5 × 3/4 = 6/20 |
| Y, Y | 2/5 × 1/4 = 2/20 |
P(at least one yellow) = 6/20 + 6/20 + 2/20 = 14/20 = 7/10
Three stages
Three coin flips: 2 × 2 × 2 = 8 paths.
P(exactly two heads): HHT, HTH, THH → 3/8.
When to use a tree diagram
- Experiments with 2–3 stages
- Different probabilities at different stages
- Dependent events
- "At least" or "exactly" questions
For many stages, trees get large; use counting rules instead.
Worked examples
Example 1: A coin and a 3-colour spinner (equal parts). How many outcomes? 2 × 3 = 6.
Example 2: P(rain) on Saturday = 0.3; on Sunday = 0.3, independent. P(rain on exactly one day)? 0.3 × 0.7 + 0.7 × 0.3 = 0.42.
Example 3: From the marble example, P(both yellow)? 1/10.
Common mistakes
- Adding along a path instead of multiplying.
- Branch probabilities from one point not adding to 1.
- Forgetting to change the second-stage probabilities for dependent events.
Try it yourself
- Draw a tree for flipping a coin then rolling a die. How many outcomes?
- A test has 2 true/false questions. If you guess, P(both right)?
- Bag: 4 red, 1 blue. Draw 2 without replacement. P(both red)?
- Three children in a family (boy/girl equally likely). P(exactly one girl)?
Answers: 1) 12 2) 1/4 3) 4/5 × 3/4 = 3/5 4) 3/8
Quick recap
- A tree diagram shows each stage with branches.
- Multiply along a path; add paths for "or".
- Branch probabilities from each point add to 1.