Answer: 0.58

This is a Bayes' theorem problem where we need to find the conditional probability that a client was in age range 31-65 given that they died. **Given data:** - Age 16-20: Mortality = 0.04, Portion = 0.1 - Age 21-30: Mortality = 0.05, Portion = 0.29 - Age 31-65: Mortality = 0.10, Portion = 0.49 - Age 66-99: Mortality = 0.14, Portion = 0.12 **Let:** - B = event that a client dies - B₁ = client is in age range 16-20 - B₂ = client is in age range 21-30 - B₃ = client is in age range 31-65 - B₄ = client is in age range 66-99 **We need:** P(B₃ | B) **Using Bayes' theorem:** P(B₃ | B) = [P(B₃) × P(B | B₃)] / [P(B₁) × P(B | B₁) + P(B₂) × P(B | B₂) + P(B₃) × P(B | B₃) + P(B₄) × P(B | B₄)] **Calculations:** - 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 probability of death:** 0.004 + 0.0145 + 0.049 + 0.0168 = 0.0843 **P(B₃ | B)** = 0.049 / 0.0843 = 0.5813 ≈ 0.58 or 58% **Why other options are incorrect:** - **B (0.172):** This is approximately P(B₂ | B) = 0.0145/0.0843 ≈ 0.172 - **C (0.168):** This is approximately P(B₄ | B) = 0.0168/0.0843 ≈ 0.199, but 0.168 is actually the numerator for B₄ - **D (0.047):** This is P(B₃) × P(B | B₃) = 0.049 (rounded to 0.047), which is just the numerator, not the conditional probability

Author: Nikitesh Somanthe