Tree Diagrams & Conditional Probability
Tree Diagrams & Conditional Probability
A tree diagram is a visual way to show all possible outcomes and calculate probabilities, especially useful for two or more successive events.
Drawing Tree Diagrams
Each branch represents a possible outcome. Write the probability on each branch. At each point, the branches must sum to 1.
Rules:
- Multiply along the branches to find P(A AND B)
- Add the results of different paths to find P(A OR B)
Worked Example: A bag contains 3 red and 5 blue beads. A bead is picked, its colour noted, and it is replaced. A second bead is then picked. Find the probability of getting exactly one red.
Draw two sets of branches:
First pick: P(R) = 3/8, P(B) = 5/8
Second pick (with replacement — same probabilities): P(R) = 3/8, P(B) = 5/8
Paths for exactly one red:
- Red then Blue: 3/8 × 5/8 = 15/64
- Blue then Red: 5/8 × 3/8 = 15/64
P(exactly one red) = 15/64 + 15/64 = 30/64 = 15/32
Without Replacement
When items are NOT replaced, the probabilities change for the second pick.
Worked Example: Same bag (3 red, 5 blue), but WITHOUT replacement. Find P(both red).
First pick: P(R) = 3/8
Second pick (one red removed, 7 left): P(R) = 2/7
P(both red) = 3/8 × 2/7 = 6/56 = 3/28
Full tree for without replacement:
First pick → Second pick:
- R (3/8) → R (2/7) = 6/56
- R (3/8) → B (5/7) = 15/56
- B (5/8) → R (3/7) = 15/56
- B (5/8) → B (4/7) = 20/56
Check: 6 + 15 + 15 + 20 = 56 = 56/56 = 1 ✓
Conditional Probability (Higher)
Conditional probability is the probability of an event given that another event has already happened. Written as P(A | B) — the probability of A given B.
Worked Example: 100 students were surveyed. 40 are girls. 15 girls and 25 boys passed a test. Find the probability that a randomly selected student passed, given that they are a girl.
- P(pass | girl) = 15/40 = 3/8
We only consider the 40 girls, not all 100 students.
Using Two-Way Tables for Conditional Probability
| Pass | Fail | Total | |
|---|---|---|---|
| Girl | 15 | 25 | 40 |
| Boy | 25 | 35 | 60 |
| Total | 40 | 60 | 100 |
- P(girl | pass) = 15/40 = 3/8 (out of those who passed, what fraction are girls?)
- P(pass | boy) = 25/60 = 5/12
Three Events (Higher)
Tree diagrams can extend to three or more events. The same rules apply — multiply along branches, add across paths.
Worked Example: The probability of rain on any day is 0.4, independently. Find P(rain on exactly 2 out of 3 days).
Paths with exactly 2 rain days (R = rain, D = dry):
- RRD: 0.4 × 0.4 × 0.6 = 0.096
- RDR: 0.4 × 0.6 × 0.4 = 0.096
- DRR: 0.6 × 0.4 × 0.4 = 0.096
P(exactly 2) = 3 × 0.096 = 0.288
Exam Tips
- Always check that branches from each node sum to 1
- "At least one" is easiest to find as 1 − P(none)
- For conditional probability, carefully identify which group you are restricting to — the "given" part becomes your new total
- Without replacement changes the denominator AND possibly the numerator on the second pick