Question

Let A and B represent events such that P(A) = 0.6, P(B) = 0.4, and P(A ∪ B) = 0.76. Compute: (a) P(A ∩ B) (b) P(Ac ∪ B) (c) P(A ∩ Bc ) (d) Are events A and B mutually exclusive? Are they independent? Explain by citing the definitions of mutual exclusivity and independence.

Answer 1

a) P(A B) = P(A) + P(B) - P(A U B)

= 0.6 + 0.4 - 0.76

= 0.24

A Venn Diagram can be used to answer the remaining parts of the question

b) P(A^c U B) = 0.24 + 0.16 + 0.24

= 0.64

c) P(A B^c) = 0.36

d) A and B are mutually exclusive if P(A B) = 0

Here, P(A B) 0

Hence, A and B are not mutually exclusive.

A and B are independent if P(A) x P(B) = P(A B)

P(A) x P(B) = 0.6 x 0.4 = 0.24

P(A B) = 0.24

P(A) x P(B) = P(A B)

Therefore, A and B are independent.