Answer: 0.047

This is a Bayes' theorem problem where we need to find the conditional probability that an insured person was in the 16-20 age group given that they died. **Step-by-step calculation:** 1. **Define events:** - B = Event of death - B₁ = Event the insured's age is 16-20 - B₂ = Event the insured's age is 21-30 - B₃ = Event the insured's age is 31-65 - B₄ = Event the insured's age is 66-99 2. **Given probabilities:** - P(B₁) = 0.1 (portion of company's insured persons aged 16-20) - P(B₂) = 0.29 - P(B₃) = 0.49 - P(B₄) = 0.12 - P(B|B₁) = 0.04 (mortality rate for age 16-20) - P(B|B₂) = 0.05 - P(B|B₃) = 0.10 - P(B|B₄) = 0.14 3. **Apply Bayes' theorem:** P(B₁|B) = [P(B₁) × P(B|B₁)] / [P(B₁) × P(B|B₁) + P(B₂) × P(B|B₂) + P(B₃) × P(B|B₃) + P(B₄) × P(B|B₄)] 4. **Calculate numerator:** P(B₁) × P(B|B₁) = 0.1 × 0.04 = 0.004 5. **Calculate denominator:** - P(B₁) × P(B|B₁) = 0.1 × 0.04 = 0.004 - 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 - Total = 0.004 + 0.0145 + 0.049 + 0.0168 = 0.0843 6. **Final calculation:** P(B₁|B) = 0.004 / 0.0843 ≈ 0.04745 ≈ 0.047 (rounded to three decimal places) **Interpretation:** Given that a randomly selected insured person has died, there is approximately a 4.7% probability that they were in the 16-20 age group.

Author: Nikitesh Somanthe