Tree diagram structure
How tree diagrams work: Branches show all outcomes. Label each branch with probability. Path probability: multiply along branches.
Worked example
Bag: 3 red, 2 blue. Draw 2 (no replace). Show tree and find P(2 red).
Solution
- Branch 1: First red (3/5)
- Sub-branch: Second red given first red (2/4)
- Path prob: (3/5)×(2/4)=6/20=3/10
- Branch 2: First blue (2/5) leads to second outcomes
Final answer
P(RR)=3/10. Multiply along red path.
Multi-stage experiments
Stages: Each branch level represents one stage. Second stage branches depend on first outcome.
Worked example
Spinner spun twice: P(red)=0.4, P(blue)=0.6. Find all outcomes and probabilities.
Solution
- RR: 0.4×0.4=0.16
- RB: 0.4×0.6=0.24
- BR: 0.6×0.4=0.24
- BB: 0.6×0.6=0.36
- Total: 0.16+0.24+0.24+0.36=1 ✓
Final answer
All paths shown, sum=1.
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Conditional probability from trees
Reading conditional prob: P(B|A) is the probability on second-stage branch GIVEN first stage was A.
Worked example
Factory: Machine A makes 60% of items, 2% defective. Machine B makes 40%, 3% defective. Find P(defective|Machine A).
Solution
- Tree: First stage: A (0.6) or B (0.4)
- From A: defective (0.02) or good (0.98)
- P(defective|A)=0.02 (second-level branch)
Final answer
0.02. Branch probability in conditional state.
Combining outcomes
Worked example
From defective example: Find P(defective).
Solution
- P(defective)=P(D|A)×P(A)+P(D|B)×P(B)
- =(0.02)×(0.6)+(0.03)×(0.4)
- =0.012+0.012=0.024
Final answer
P(defective)=0.024. Add paths leading to defective.