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?
Explanation:
Explanation
This is a Bayes' theorem problem where we need to find the conditional probability 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
- 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)] / [Total probability of death]
Total probability of death = 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)
= (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
Therefore, given that a patient died, there is approximately a 36% probability that they were in stage 4 cancer.