# Help with Probability question.

1. Feb 17, 2012

### Kiziaru

So I've been trying to do probability problems to take the P/1 Exam, and I've come across a problem that I don't quite understand.

Problem:

An insurance company estimates that 40% of policyholders who have only an auto policy
will renew next year and 60% of policyholders who have only a homeowners policy will
renew next year. The company estimates that 80% of policyholders who have both an
auto and a homeowners policy will renew at least one of those policies next year.
Company records show that 65% of policyholders have an auto policy, 50% of
policyholders have a homeowners policy, and 15% of policyholders have both an auto
and a homeowners policy. Using the company’s estimates, calculate the percentage of
policyholders that will renew at least one policy next year.

Attempt at solution:

Pr(A ∩ H(compliment)) = Pr(A - H) = Pr(A) - Pr(H)

Pr(A) = .65
Pr(H) = .50

.65 - .50 = .15

However, the solution claims that Pr(A - H) = Pr(A-(A ∩ H)) = .50. How is that possible? How did they derive that?

Edit: Disregard, I understand it now. I had to apply the Pr(A U B) = Pr(A) + Pr(B) - Pr(A ∩ B) rule.

Last edited: Feb 17, 2012