A company sells two types of life insurance policies (P and Q) and one type of health insurance policy. A survey of potential customers revealed the following:

1. No survey participant wanted to purchase both life policies.
2. Twice as many survey participants wanted to purchase life policy P as life policy Q.
3. 45% of survey participants wanted to purchase the health policy.
4. 18% of survey participants wanted to purchase only the health policy.
5. The event that a survey participant wanted to purchase the health policy was independent of the event that a survey participant wanted to purchase a life policy.

Calculate the probability that a randomly selected survey participant wanted to purchase exactly one policy.

• 0.51
• 0.60
• 0.69
• 0.73
• 0.78

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

Let A = event that person wants life policy P

B = event that person wants life policy Q

C = event that person wants the health policy

and let a, b, c, d be the probabilities of the regions as shown.

1. is reflected by no intersection of A and B
2. is reflected by the 0.18 in the diagram
3. implies a + b = 2(c + d)
4. implies b + c + 0.18 = 0.45 or b + c = 0.27
5. implies P([A or B] and C) = P(A or B)P(C) or b + c = (a + b + c + d)(0.45)

So 0.27 = (a + d + 0.27)(0.45) and then a + d = 0.33. The desired probability is a + 0.18 + d = 0.33 + 0.18 = 0.51.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.