Worked probability tree · printed probabilities
Three colour draws with replacement
Read the conditional branches and the product for each complete outcome. These values stay fixed to the source example.
Expand a first-draw branch to read its nine complete paths. All 27 ordered outcomes are retained.
Start · probability 1
Open a branch to follow its outcomes. Outcome percentages are for complete paths. Open an outcome to see its calculation.
W₁ · White0.5 · 9 outcomes
W₁ · White
W₂ · White0.5 · 3 outcomes
W₁ · White → W₂ · White
W₃ · WhitePath 12.5%
W₁ · White → W₂ · White → W₃ · White- Branch weights
- 0.5 × 0.5 × 0.5
- Complete path probability
- 1/8= 0.125 · 12.5%
R₃ · RedPath 7.5%
W₁ · White → W₂ · White → R₃ · Red- Branch weights
- 0.5 × 0.5 × 0.3
- Complete path probability
- 3/40= 0.075 · 7.5%
G₃ · GreenPath 5%
W₁ · White → W₂ · White → G₃ · Green- Branch weights
- 0.5 × 0.5 × 0.2
- Complete path probability
- 1/20= 0.05 · 5%
R₂ · Red0.3 · 3 outcomes
W₁ · White → R₂ · Red
W₃ · WhitePath 7.5%
W₁ · White → R₂ · Red → W₃ · White- Branch weights
- 0.5 × 0.3 × 0.5
- Complete path probability
- 3/40= 0.075 · 7.5%
R₃ · RedPath 4.5%
W₁ · White → R₂ · Red → R₃ · Red- Branch weights
- 0.5 × 0.3 × 0.3
- Complete path probability
- 9/200= 0.045 · 4.5%
G₃ · GreenPath 3% · All different
W₁ · White → R₂ · Red → G₃ · Green- Branch weights
- 0.5 × 0.3 × 0.2
- Complete path probability
- 3/100= 0.03 · 3%
G₂ · Green0.2 · 3 outcomes
W₁ · White → G₂ · Green
W₃ · WhitePath 5%
W₁ · White → G₂ · Green → W₃ · White- Branch weights
- 0.5 × 0.2 × 0.5
- Complete path probability
- 1/20= 0.05 · 5%
R₃ · RedPath 3% · All different
W₁ · White → G₂ · Green → R₃ · Red- Branch weights
- 0.5 × 0.2 × 0.3
- Complete path probability
- 3/100= 0.03 · 3%
G₃ · GreenPath 2%
W₁ · White → G₂ · Green → G₃ · Green- Branch weights
- 0.5 × 0.2 × 0.2
- Complete path probability
- 1/50= 0.02 · 2%
R₁ · Red0.3 · 9 outcomes
R₁ · Red
W₂ · White0.5 · 3 outcomes
R₁ · Red → W₂ · White
W₃ · WhitePath 7.5%
R₁ · Red → W₂ · White → W₃ · White- Branch weights
- 0.3 × 0.5 × 0.5
- Complete path probability
- 3/40= 0.075 · 7.5%
R₃ · RedPath 4.5%
R₁ · Red → W₂ · White → R₃ · Red- Branch weights
- 0.3 × 0.5 × 0.3
- Complete path probability
- 9/200= 0.045 · 4.5%
G₃ · GreenPath 3% · All different
R₁ · Red → W₂ · White → G₃ · Green- Branch weights
- 0.3 × 0.5 × 0.2
- Complete path probability
- 3/100= 0.03 · 3%
R₂ · Red0.3 · 3 outcomes
R₁ · Red → R₂ · Red
W₃ · WhitePath 4.5%
R₁ · Red → R₂ · Red → W₃ · White- Branch weights
- 0.3 × 0.3 × 0.5
- Complete path probability
- 9/200= 0.045 · 4.5%
R₃ · RedPath 2.7%
R₁ · Red → R₂ · Red → R₃ · Red- Branch weights
- 0.3 × 0.3 × 0.3
- Complete path probability
- 27/1000= 0.027 · 2.7%
G₃ · GreenPath 1.8%
R₁ · Red → R₂ · Red → G₃ · Green- Branch weights
- 0.3 × 0.3 × 0.2
- Complete path probability
- 9/500= 0.018 · 1.8%
G₂ · Green0.2 · 3 outcomes
R₁ · Red → G₂ · Green
W₃ · WhitePath 3% · All different
R₁ · Red → G₂ · Green → W₃ · White- Branch weights
- 0.3 × 0.2 × 0.5
- Complete path probability
- 3/100= 0.03 · 3%
R₃ · RedPath 1.8%
R₁ · Red → G₂ · Green → R₃ · Red- Branch weights
- 0.3 × 0.2 × 0.3
- Complete path probability
- 9/500= 0.018 · 1.8%
G₃ · GreenPath 1.2%
R₁ · Red → G₂ · Green → G₃ · Green- Branch weights
- 0.3 × 0.2 × 0.2
- Complete path probability
- 3/250= 0.012 · 1.2%
G₁ · Green0.2 · 9 outcomes
G₁ · Green
W₂ · White0.5 · 3 outcomes
G₁ · Green → W₂ · White
W₃ · WhitePath 5%
G₁ · Green → W₂ · White → W₃ · White- Branch weights
- 0.2 × 0.5 × 0.5
- Complete path probability
- 1/20= 0.05 · 5%
R₃ · RedPath 3% · All different
G₁ · Green → W₂ · White → R₃ · Red- Branch weights
- 0.2 × 0.5 × 0.3
- Complete path probability
- 3/100= 0.03 · 3%
G₃ · GreenPath 2%
G₁ · Green → W₂ · White → G₃ · Green- Branch weights
- 0.2 × 0.5 × 0.2
- Complete path probability
- 1/50= 0.02 · 2%
R₂ · Red0.3 · 3 outcomes
G₁ · Green → R₂ · Red
W₃ · WhitePath 3% · All different
G₁ · Green → R₂ · Red → W₃ · White- Branch weights
- 0.2 × 0.3 × 0.5
- Complete path probability
- 3/100= 0.03 · 3%
R₃ · RedPath 1.8%
G₁ · Green → R₂ · Red → R₃ · Red- Branch weights
- 0.2 × 0.3 × 0.3
- Complete path probability
- 9/500= 0.018 · 1.8%
G₃ · GreenPath 1.2%
G₁ · Green → R₂ · Red → G₃ · Green- Branch weights
- 0.2 × 0.3 × 0.2
- Complete path probability
- 3/250= 0.012 · 1.2%
G₂ · Green0.2 · 3 outcomes
G₁ · Green → G₂ · Green
W₃ · WhitePath 2%
G₁ · Green → G₂ · Green → W₃ · White- Branch weights
- 0.2 × 0.2 × 0.5
- Complete path probability
- 1/50= 0.02 · 2%
R₃ · RedPath 1.2%
G₁ · Green → G₂ · Green → R₃ · Red- Branch weights
- 0.2 × 0.2 × 0.3
- Complete path probability
- 3/250= 0.012 · 1.2%
G₃ · GreenPath 0.8%
G₁ · Green → G₂ · Green → G₃ · Green- Branch weights
- 0.2 × 0.2 × 0.2
- Complete path probability
- 1/125= 0.008 · 0.8%
Compare outcome probabilities
| Outcome | Probability |
|---|---|
| W₁ · White → W₂ · White → W₃ · White | 1/8= 0.125 · 12.5% |
| W₁ · White → W₂ · White → R₃ · Red | 3/40= 0.075 · 7.5% |
| W₁ · White → W₂ · White → G₃ · Green | 1/20= 0.05 · 5% |
| W₁ · White → R₂ · Red → W₃ · White | 3/40= 0.075 · 7.5% |
| W₁ · White → R₂ · Red → R₃ · Red | 9/200= 0.045 · 4.5% |
| W₁ · White → R₂ · Red → G₃ · Green | 3/100= 0.03 · 3% |
| W₁ · White → G₂ · Green → W₃ · White | 1/20= 0.05 · 5% |
| W₁ · White → G₂ · Green → R₃ · Red | 3/100= 0.03 · 3% |
| W₁ · White → G₂ · Green → G₃ · Green | 1/50= 0.02 · 2% |
| R₁ · Red → W₂ · White → W₃ · White | 3/40= 0.075 · 7.5% |
| R₁ · Red → W₂ · White → R₃ · Red | 9/200= 0.045 · 4.5% |
| R₁ · Red → W₂ · White → G₃ · Green | 3/100= 0.03 · 3% |
| R₁ · Red → R₂ · Red → W₃ · White | 9/200= 0.045 · 4.5% |
| R₁ · Red → R₂ · Red → R₃ · Red | 27/1000= 0.027 · 2.7% |
| R₁ · Red → R₂ · Red → G₃ · Green | 9/500= 0.018 · 1.8% |
| R₁ · Red → G₂ · Green → W₃ · White | 3/100= 0.03 · 3% |
| R₁ · Red → G₂ · Green → R₃ · Red | 9/500= 0.018 · 1.8% |
| R₁ · Red → G₂ · Green → G₃ · Green | 3/250= 0.012 · 1.2% |
| G₁ · Green → W₂ · White → W₃ · White | 1/20= 0.05 · 5% |
| G₁ · Green → W₂ · White → R₃ · Red | 3/100= 0.03 · 3% |
| G₁ · Green → W₂ · White → G₃ · Green | 1/50= 0.02 · 2% |
| G₁ · Green → R₂ · Red → W₃ · White | 3/100= 0.03 · 3% |
| G₁ · Green → R₂ · Red → R₃ · Red | 9/500= 0.018 · 1.8% |
| G₁ · Green → R₂ · Red → G₃ · Green | 3/250= 0.012 · 1.2% |
| G₁ · Green → G₂ · Green → W₃ · White | 1/50= 0.02 · 2% |
| G₁ · Green → G₂ · Green → R₃ · Red | 3/250= 0.012 · 1.2% |
| G₁ · Green → G₂ · Green → G₃ · Green | 1/125= 0.008 · 0.8% |
| All 27 outcomes | 1= 1 · 100% |
Tree assumptions and source notes
W: white, R: red, G: green. Every split uses 0.5, 0.3 and 0.2 because each ball is replaced.
Draw indices 1, 2 and 3 identify events, not individual balls.
The six all-different paths are WRG, WGR, RWG, RGW, GWR and GRW. They are labeled All different in the tree; each has probability 0.03.
This diagram is for the with-replacement experiment in part (i), not the five-ball selection without replacement in part (ii).
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