Answer: 0.172

This is a Bayes' theorem problem where we need to find the conditional probability that a dead client was in the 21-30 age range given that they died. **Step-by-step solution:** Let: - B₁ = event that client is in age range 16-20 - B₂ = event that client is in age range 21-30 - B₃ = event that client is in age range 31-65 - B₄ = event that client is in age range 66-99 - B = event that client dies From the table: - P(B₁) = 0.1, P(B|B₁) = 0.04 - P(B₂) = 0.29, P(B|B₂) = 0.05 - P(B₃) = 0.49, P(B|B₃) = 0.10 - P(B₄) = 0.12, P(B|B₄) = 0.14 Using Bayes' theorem: P(B₂ | B) = [P(B₂) × P(B|B₂)] / [P(B₁) × P(B|B₁) + P(B₂) × P(B|B₂) + P(B₃) × P(B|B₃) + P(B₄) × P(B|B₄)] **Calculation:** Numerator: 0.29 × 0.05 = 0.0145 Denominator: - (0.1 × 0.04) = 0.004 - (0.29 × 0.05) = 0.0145 - (0.49 × 0.10) = 0.049 - (0.12 × 0.14) = 0.0168 Total denominator = 0.004 + 0.0145 + 0.049 + 0.0168 = 0.0843 P(B₂ | B) = 0.0145 / 0.0843 = 0.172 ≈ 17.2% **Key insight:** This is a classic application of Bayes' theorem for conditional probability, where we're finding the probability of being in a particular age group given that death has occurred. The answer 0.172 corresponds to option A.

Author: Nikitesh Somanthe