Answer-first summary for fast verification
Answer: 57%
The correct answer to the question is D, which is 57%. The explanation for this can be derived from the given probabilities and the relationships between the events A (having an auto policy) and H (having a homeowner policy). Here's a comprehensive breakdown: 1. **Understanding the Events**: - A: Policyholder has an auto policy. - H: Policyholder has a homeowner policy. - A ∩ H: Policyholder has both an auto and a homeowner policy. - A°: Policyholder has only an auto policy (A without H). - H°: Policyholder has only a homeowner policy (H without A). 2. **Given Probabilities**: - P(A ∩ H) = 0.20 (20% have both policies). - P(A) = 0.70 (70% have an auto policy). - P(H) = 0.50 (50% have a homeowner policy). 3. **Calculating Complementary Probabilities**: - P(A°) = P(A) - P(A ∩ H) = 0.70 - 0.20 = 0.50 (50% have only an auto policy). - P(H°) = P(H) - P(A ∩ H) = 0.50 - 0.20 = 0.30 (30% have only a homeowner policy). 4. **Renewal Probabilities**: - 40% of those with only an auto policy will renew (P(A°) * 0.40). - 70% of those with only a homeowner policy will renew (P(H°) * 0.70). - 80% of those with both policies will renew at least one (P(A ∩ H) * 0.80). 5. **Calculating the Total Renewal Rate**: - Total renewal rate = (0.50 * 0.40) + (0.30 * 0.70) + (0.20 * 0.80) = 0.20 + 0.21 + 0.16 = 0.57 or 57%. This calculation shows that 57% of policyholders will renew at least one policy next year, which is the correct answer provided in the file content.
Author: LeetQuiz Editorial Team
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In the context of an insurance company's projections, it is anticipated that 40% of policyholders who have only an auto policy will renew it in the upcoming year. Additionally, 70% of those with solely a homeowner policy are expected to renew. Furthermore, 80% of individuals holding both types of policies are estimated to renew at least one policy. Based on the company’s records, 70% of policyholders have an auto policy, 50% have a homeowner policy, and 20% possess both policies. Given these statistics, what proportion of the entire pool of policyholders is projected to renew at least one policy in the coming year?
A
29%
B
41%
C
53%
D
57%