Question

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:

i) No survey participant wanted to purchase both life policies.

ii) Twice as many survey participants wanted to purchase life policy P as life policy Q.

iii) 45% of survey participants wanted to purchase the health policy.

iv) 18% of survey participants wanted to purchase only the health policy.

v) 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.

The answer is 0.51. If someone could show the process to solve and explain more about how to handle the last fact (v), it would be very helpful!

Answer 1

Here is the venn diagram for all the three events:

Now, from the given statements:

(i) No survey participants wanted to purchase both life policies. Intersection of P and Q is zero. Hence, b = 0, g = 0

(ii) Twice as many wanted to purchase P than Q:

(a + d) = 2*(c+f)

(iii) e + f + g + d = 0.45

Hence, e + f + d = 0.45

(iv) e = 0.18

From iii and iv,

f + d = 0.27

Also, sum of all probabilities = 1

Hence, a + b + c + d + e + f + g = 1

OR a + 0 + c + d + 0.18 + f = 1

a + c + d + f = 0.82

a + c + 0.27 = 0.82

Hence, a + c = 0.55

Probability that the selected participant wanted to purchase exactly one policy = a + c + e = 0.55 + 0.18 = 0.73

Hence, 0.73 is the correct answer. Please check your answer. The last fact is just that the two events are independent of each other, it is an assumption. You don't have to worry about the last fact.

The answer could also be tallied using logical reasoning:

45% wants to purchase the health policy while 18% want to purchase only the health policy. Hence, (45 - 18)% = 27% want to purchase Life + Health Policy. Hence, the remaining people of the population (100 - 27 - 18) = 55% will only have a life policy.Therefore, the %age of participants only having one policy = 55% + 18% = 73%.