Probability for Actuaries Question

[JosCC]

Hi all! I was solving this question, and got the wrong answer. It is a bit long so I apologize.

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.

My solution:
I drew a vinn diagram to present the 65%, 50% and 15%
A∩H=15%
I denoted A=Auto, H=Homeowner insurance.
I wanted to find the individual percentage of A and H
so I subtracted 65%-A∩H=50% for A
then did the same for Homeowners, 50%-A∩H=35%
The question then asks for at least one policy next year. So what I did was use the inclusion-exclusion principle
n(A U B)=n(A)-n(B)-n(A∩B)
Hence,
n(A U H)= .60*(.50)+.4(.35)-A∩H (15%) to get the final answer of 29%.
Where did I go wrong?

 

[Steven G]

You have three separate regions.
Policy owners who only have Auto insurance is 50%--40% will renew-----------------------------------------------------20%
Policy owners who only have homeowners insurance is 35%---60% will renew-------------------------------------------21%
Policy owners who have both auto and homeowners insurance is 15%---80% will renew al least one policy----------12%

This is a hard test which you are preparing for and need to be able to do these type of problems. I hope that I did!