Let A and B be two events such that `p( bar AuuB)=1/6, p(AnnB)=1/4` and `p( bar A)=1/4` , where ` bar A` stands for the complement of the event A. The

Question

Let A and B be two events such that p( ˉ A ∪B)= 1 6 ,p(A∩B)= 1 4 and p( ˉ A )= 1 4 , where ˉ A stands for the complement of the event A. Then the events A and B are (1) mutually exclusive and independent (2) equally likely but not independent (3) independent but not equally likely (4) independent and equally likely A. mutually exclusive and independent B. independent but not equally likely C. equally likely but not independent D. equally likely and mutually exclusive Select the correct answer from above options

Answer 1

Correct Answer - B We have, P ( A ∪ B ) = 1 4 ≠ 0 P So, A and B are not mutually exclusive events. Now, P ( ¯¯¯¯¯¯¯¯¯¯ A ∪ B ) = 1 6 P ⇒ 1 − P ( A ∪ B ) = 1 6 ⇒ ⇒ P ( A ∪ B ) = 5 6 ⇒ ⇒ P ( A ) + P ( B ) − P ( A ∩ B ) = 5 6 ⇒ ⇒ 3 4 + P ( B ) − 1 4 = 5 6 ⇒ P ( B ) = 1 3 ⇒ Clearly, P ( A ∪ B ) = 1 4 = 3 4 × 1 3 = P ( A ) P ( B ) P So, A and B are independent. Also, P ( A ) ≠ P ( B ) P . So, A and B are not equally likely.