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Answer: 57%
## Explanation Let's define the events: - **A** = event that a policyholder has an auto policy - **H** = event that a policyholder has a homeowner policy **Given probabilities:** - P(A) = 0.70 (70% have auto policy) - P(H) = 0.50 (50% have homeowner policy) - P(A ∩ H) = 0.20 (20% have both policies) **Calculate probabilities for each group:** - **Only auto policy**: P(A ∩ Hᶜ) = P(A) - P(A ∩ H) = 0.70 - 0.20 = 0.50 - **Only homeowner policy**: P(Aᶜ ∩ H) = P(H) - P(A ∩ H) = 0.50 - 0.20 = 0.30 - **Both policies**: P(A ∩ H) = 0.20 **Renewal rates:** - Only auto policy: 40% renewal rate - Only homeowner policy: 70% renewal rate - Both policies: 80% renewal rate (at least one policy) **Calculate expected renewal percentage:** \[\text{Renewal percentage} = 0.40 \times P(A \cap H^c) + 0.70 \times P(A^c \cap H) + 0.80 \times P(A \cap H)\] \[= 0.40 \times 0.50 + 0.70 \times 0.30 + 0.80 \times 0.20\] \[= 0.20 + 0.21 + 0.16 = 0.57 = 57\%\] Therefore, **57%** of policyholders are expected to renew at least one policy next year.
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An insurance company estimates that 40% of policyholders who have only an auto policy will renew next year, and 70% of policyholders who have only a homeowner policy will renew next year. The company estimates that 80% of policyholders who have both an auto and a homeowner policy will renew at least one of those policies next year. Company records show that 70% of policyholders have an auto policy, 50% of policyholders have a homeowner policy, and 20% of policyholders have both an auto and a homeowner policy. Using the company's estimates, what is the percentage of policyholders that will renew at least one policy next year?
A
29%
B
41%
C
53%
D
57%