Let A and B be independent events. Use the definition of

QUESTION:

Let A and B be independent events. Use the definition of independence to prove the following: (a) The events A and Be are independent. (b) The events AC and BC are independent. Solution. (a) The event A is the union of the disjoint events An BC and An B. Using the additivity axiom and the independence of A and B, we obtain P(A) = P(A n B) + P(A n BC) = P(A)P(B) + P(A n BC). It follows that P(A n Be) = P(A) (1 - P(B)) = P(A)P(BC). so A and BC are independent. (b) Apply the result of part (a) twice: first on A and B. then on BC and A.