Answer: 0.0475

## Solution Explanation This is a **Bayes' Theorem** problem where we need to find the conditional probability that a deceased client was in the 16-20 age group. ### Step 1: Define Events - **B** = Event of death - **B₁** = Event the insured's age is in the range 16-20 - **B₂** = Event the insured's age is in the range 21-30 - **B₃** = Event the insured's age is in the range 31-65 - **B₄** = Event the insured's age is in the range 66-99 ### Step 2: Apply Bayes' Theorem We want to find P(B₁|B): $$P(B_1|B) = \frac{P(B_1) \cdot P(B|B_1)}{P(B_1) \cdot P(B|B_1) + P(B_2) \cdot P(B|B_2) + P(B_3) \cdot P(B|B_3) + P(B_4) \cdot P(B|B_4)}$$ ### Step 3: Calculate Numerator - P(B₁) = 0.10 (portion of company's insured persons) - P(B|B₁) = 0.04 (mortality rate for age 16-20) - Numerator = 0.10 × 0.04 = 0.004 ### Step 4: Calculate Denominator - P(B₂) × P(B|B₂) = 0.29 × 0.05 = 0.0145 - P(B₃) × P(B|B₃) = 0.49 × 0.10 = 0.049 - P(B₄) × P(B|B₄) = 0.12 × 0.14 = 0.0168 - Denominator = 0.004 + 0.0145 + 0.049 + 0.0168 = 0.0843 ### Step 5: Final Calculation $$P(B_1|B) = \frac{0.004}{0.0843} = 0.04745 \approx 4.75\%$$ ### Step 6: Interpretation The probability that a randomly selected deceased client was age 16-20 is approximately **4.75%**, which matches option D (0.0475). This makes intuitive sense because although the 16-20 age group has the lowest mortality rate (0.04), they also represent only 10% of the total insured population, resulting in a relatively small conditional probability.

Author: Tanishq Prabhu