Financial Risk Manager Part 1
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Upon arrival at a cancer treatment center, patients are categorized into one of four stages namely: stage 1, stage 2, stage 3, and stage 4. In the past year,
i. 10% of patients arriving were in stage 1
ii. 40% of patients arriving were in stage 2
iii. 30% of patients arriving were in stage 3
iv. The rest of the patients were in stage 4
v. 10% of stage 1 patients died
vi. 20% of stage 2 patients died
vii. 30% of stage 3 patients died
viii. 50% of stage 4 patient died
Given that a patient survived, what is the probability that the patient was in stage 4 upon arrival?
Other
Community
NNikitesh
A
12.99%
B
13.88%
C
15.22%
D
13.22%
Explanation:
This is a conditional probability problem using Bayes' theorem. We need to find P(C₄ | D') where D' denotes survival.
Given probabilities:
- P(C₁) = 0.10 (stage 1)
- P(C₂) = 0.40 (stage 2)
- P(C₃) = 0.30 (stage 3)
- P(C₄) = 1 - (0.10 + 0.40 + 0.30) = 0.20 (stage 4)
Death probabilities:
- P(D|C₁) = 0.10 → P(D'|C₁) = 0.90
- P(D|C₂) = 0.20 → P(D'|C₂) = 0.80
- P(D|C₃) = 0.30 → P(D'|C₃) = 0.70
- P(D|C₄) = 0.50 → P(D'|C₄) = 0.50
Applying Bayes' theorem: P(C₄ | D') = [P(C₄) × P(D'|C₄)] / [P(C₁) × P(D'|C₁) + P(C₂) × P(D'|C₂) + P(C₃) × P(D'|C₃) + P(C₄) × P(D'|C₄)]
Calculation: Numerator: 0.20 × 0.50 = 0.10
Denominator:
- Stage 1: 0.10 × 0.90 = 0.09
- Stage 2: 0.40 × 0.80 = 0.32
- Stage 3: 0.30 × 0.70 = 0.21
- Stage 4: 0.20 × 0.50 = 0.10 Total denominator = 0.09 + 0.32 + 0.21 + 0.10 = 0.72
P(C₄ | D') = 0.10 / 0.72 = 0.138888... ≈ 13.88%
Verification: The total probability of survival P(D') = 0.72, which makes sense as it's the weighted average of survival rates across all stages. The probability that a surviving patient was in stage 4 is 13.88%, which corresponds to option B.
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