Question

consider the following statements concerning the probabilities of two events, A and B: P(A U B)= 0.85, P(A/B)= 0.54, P(B)= 0.5 . Determine whether the events A and B are: (a) mutually exclusive, (b) independent

Answer 1

We have

P(A U B)= 0.85, P(A/B)= 0.54, P(B)= 0.5

By the formula of the conditional probability:

Also, by addition law for any two events A and B,

P(A∪B)=P(A)+P(B)−P(A∩B)

P(A)=P(A∪B) - P(B) +P(A∩B)

P(A)= 0.85 - 0.5 + 0.27

= 0.62

a) Now, A and B are mutually exclusive events,if P(A∩B) = 0 i.e. P(A∪B)=P(A)+P(B).

Since P(A∩B) = 0.27 , hence A and B are not mutually exclusive events.

b) A and B are independent events, if P(A∩B) = P(A)P(B).

Since P(A∩B) = 0.27

& P(A)P(B) = 0.5 *0.62

= 0.31 P(A∩B)

Hence A and B are not independent events.