Tree Diagrams - Visualising Multi-Step Probability
A tree diagram is a visual tool for listing all outcomes of a multi-step probability experiment and calculating probabilities by multiplying along branches. Tree diagrams make it easy to see every possibility at once and to apply both the multiplication rule and the addition rule systematically.
How a Tree Diagram Works
- Each branch represents one possible outcome at a given step.
- The probability is written on the branch.
- To find the probability of a path through the tree, multiply the probabilities along its branches.
- To find the probability of an event made up of several paths, add the path probabilities.
The Two Rules
Along branches (AND): Multiply probabilities.
Between paths (OR): Add probabilities.
Worked Examples
| Toss 1 | Toss 2 | Outcome | Probability |
|---|---|---|---|
| H (1/2) | H (1/2) | HH | 1/2 × 1/2 = 1/4 |
| H (1/2) | T (1/2) | HT | 1/2 × 1/2 = 1/4 |
| T (1/2) | H (1/2) | TH | 1/2 × 1/2 = 1/4 |
| T (1/2) | T (1/2) | TT | 1/2 × 1/2 = 1/4 |
P(exactly one head) = P(HT) + P(TH) = 1/4 + 1/4 = 1/2. Check: all paths sum to 1. ✓
| 1st draw | 2nd draw | Outcome | Probability |
|---|---|---|---|
| R (3/5) | R (2/4) | RR | 3/5 × 2/4 = 6/20 |
| R (3/5) | B (2/4) | RB | 3/5 × 2/4 = 6/20 |
| B (2/5) | R (3/4) | BR | 2/5 × 3/4 = 6/20 |
| B (2/5) | B (1/4) | BB | 2/5 × 1/4 = 2/20 |
P(both same) = P(RR) + P(BB) = 6/20 + 2/20 = 8/20 = 2/5. Check: 6+6+6+2 = 20. ✓
P(rain and late) = 0.3 × 0.7 = 0.21.
P(no rain and late) = 0.7 × 0.2 = 0.14.
P(bus is late) = 0.21 + 0.14 = 0.35.
Key Takeaways
- Write probabilities on the branches, outcomes at the end of each path.
- Multiply along branches (AND rule) to find each path's probability.
- Add path probabilities (OR rule) when an event can happen via more than one path.
- All end-of-path probabilities must sum to 1 – use this as a check.
Practice Questions
- Draw a tree diagram for spinning a spinner (Red 1/3, Blue 2/3) twice. Find P(both red) and P(at least one blue).
- A bag has 4 yellow and 6 green balls. Two are drawn with replacement. Draw a tree diagram and find P(one of each colour).
- A bag has 4 yellow and 6 green balls. Two are drawn WITHOUT replacement. Find P(both yellow).
- P(sunny) = 0.6. If sunny, P(picnic) = 0.9. If not sunny, P(picnic) = 0.1. Find P(picnic).
- A student takes two tests. P(pass test 1) = 0.8. If they pass test 1, P(pass test 2) = 0.9; if they fail test 1, P(pass test 2) = 0.5. Find P(pass both) and P(pass exactly one).
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