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 the patient died, what is the probability that the patient was in stage 4 cancer?
Other
Community
NNikitesh
A
0.86
B
0.1
C
0.36
D
0.5
Explanation:
Explanation
This is a classic Bayes' Theorem problem. We need to find the probability that a patient was in stage 4 given that they died: P(Stage 4 | Died).
Given probabilities:
- P(Stage 1) = 0.10
- P(Stage 2) = 0.40
- P(Stage 3) = 0.30
- P(Stage 4) = 1 - (0.10 + 0.40 + 0.30) = 0.20
Conditional probabilities of death given stage:
- P(Died | Stage 1) = 0.10
- P(Died | Stage 2) = 0.20
- P(Died | Stage 3) = 0.30
- P(Died | Stage 4) = 0.50
Applying Bayes' Theorem:
P(Stage 4 | Died) = [P(Stage 4) × P(Died | Stage 4)] / P(Died)
Where P(Died) = total probability of death: P(Died) = P(Stage 1)×P(Died|Stage 1) + P(Stage 2)×P(Died|Stage 2) + P(Stage 3)×P(Died|Stage 3) + P(Stage 4)×P(Died|Stage 4)
Calculation:
P(Died) = (0.10 × 0.10) + (0.40 × 0.20) + (0.30 × 0.30) + (0.20 × 0.50) = 0.01 + 0.08 + 0.09 + 0.10 = 0.28
P(Stage 4 | Died) = (0.20 × 0.50) / 0.28 = 0.10 / 0.28 = 0.3571 ≈ 0.36 or 36%
Interpretation:
Given that a patient died, there's a 36% probability they were in stage 4 cancer. This makes sense because although stage 4 has the highest mortality rate (50%), it represents only 20% of the patient population. The other stages contribute significantly to the total deaths despite their lower individual mortality rates.
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