Answer-first summary for fast verification
Answer: 9%
This is a Bayes' theorem application problem. We need to find P(U|C) - the probability that a policyholder is from the ultra preferred tier given that they have a claim. **Given probabilities:** - P(S) = 0.25 (Standard tier) - P(P) = 0.50 (Preferred tier) - P(U) = 0.25 (Ultra preferred tier) - P(C|S) = 0.10 (Claim given Standard) - P(C|P) = 0.05 (Claim given Preferred) - P(C|U) = 0.02 (Claim given Ultra preferred) **Step 1: Calculate total probability of a claim P(C)** P(C) = P(C|S)×P(S) + P(C|P)×P(P) + P(C|U)×P(U) P(C) = 0.10×0.25 + 0.05×0.50 + 0.02×0.25 P(C) = 0.025 + 0.025 + 0.005 = 0.055 **Step 2: Apply Bayes' theorem** P(U|C) = [P(C|U)×P(U)] / P(C) P(U|C) = (0.02×0.25) / 0.055 P(U|C) = 0.005 / 0.055 = 0.0909 ≈ 9% **Verification:** The calculation shows that even though ultra preferred policyholders have the lowest claim rate (2%), they represent 25% of the population, so their contribution to the total claims is 0.005 out of 0.055 total claims, which is about 9%.
Author: Nikitesh Somanthe
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An insurance company classifies its policyholders into three tiers – standard, preferred and ultra preferred with a 25%/50%/25% distribution. The chance of a policyholder in the standard tier having a claim is 10%, in the preferred tier it is 5% and in the ultra preferred tier it is 2%. Given a policyholder has a claim, what is the probability they came from the ultra preferred tier?
A
5%
B
7%
C
9%
D
11%