Answer: 9%

## Explanation This is a classic Bayes' Theorem application. We need to find the probability that a policyholder came from the ultra preferred tier given that they have a claim: $P(U|C)$. ### Given Information: - Distribution: Standard (25%), Preferred (50%), Ultra Preferred (25%) - Claim probabilities: - Standard: 10% (0.10) - Preferred: 5% (0.05) - Ultra Preferred: 2% (0.02) ### Using Bayes' Theorem: $$P(U|C) = \frac{P(C|U)P(U)}{P(C)}$$ ### Step 1: Calculate $P(C)$ - Overall probability of a claim $$P(C) = P(C|S)P(S) + P(C|P)P(P) + P(C|U)P(U)$$ $$P(C) = (0.10 \times 0.25) + (0.05 \times 0.50) + (0.02 \times 0.25)$$ $$P(C) = 0.025 + 0.025 + 0.005 = 0.055$$ ### Step 2: Calculate $P(U|C)$ $$P(U|C) = \frac{P(C|U)P(U)}{P(C)} = \frac{0.02 \times 0.25}{0.055} = \frac{0.005}{0.055} = 0.0909 \approx 9\%$$ ### Verification: The result makes intuitive sense - although ultra preferred policyholders have the lowest claim rate (2%), they represent a significant portion of the population (25%), and their low claim rate combined with their population proportion gives them about 9% of all claims.

Author: Tanishq Prabhu