Question

Ink Inc., a publishing firm, offers its 755 employees a choice of benefits: 314 enrolled in the firm’s vision plan, 241 enrolled in the firm’s disability insurance, and 225 enrolled in both the vision plan and disability insurance.

(a) Draw a Venn Diagram (with probabilities) that illustrates this situation.

(b) What is the probability that someone did not enroll in either benefit?

(c) Are the events ‘enrolling in the firm’s vision plan’ and ‘enrolling in the firm’s disability insurance’ independent?

(d) Are the events ‘enrolling in the firm’s vision plan’ and ‘enrolling in the firm’s disability insurance’ mutually exclusive?

(e) Give an example of two events (in this described situation) that are mutually exclusive.

Answer 1

(a) Draw a Venn Diagram (with probabilities) that illustrates this situation.

(b) What is the probability that someone did not enroll in either benefit?

probability that someone did not enroll in either benefit = 1 - P(AUB)

P(AUB) = P(A) + P(B) - P(A^B)

P(AUB) = 0.42 +0.32 - 0.3

P(AUB) = 0.44

probability that someone did not enroll in either benefit = 1 - P(AUB)

probability that someone did not enroll in either benefit = 1 - 0.44

probability that someone did not enroll in either benefit = 0.56

(c) Are the events ‘enrolling in the firm’s vision plan’ and ‘enrolling in the firm’s disability insurance’ independent?

P(A)*P(B) = 0.42*0.32 = 0.1344

P(A^B) = 0.3

P(A)*P(B) is not equal to P(A^B) . hence, not independent

(d) Are the events ‘enrolling in the firm’s vision plan’ and ‘enrolling in the firm’s disability insurance’ mutually exclusive?

No, Since,  P(A^B) > 0

events ‘enrolling in the firm’s vision plan’ and ‘enrolling in the firm’s disability insurance’ are not mutually exclusive

(e) Give an example of two events (in this described situation) that are mutually exclusive.

Someone, enroll in either of the plan and not enrolling in any plan are mutually exclusive events