Complement and Independent events.

Question

Show that if A and B are independent events, then the pairs A and B', A' and B, A' and B' are also independent.

How do I prove it?

Answer 1

If A and B independent, P(A and B) = P(A)P(B). Since B and B' are mutually exclusive, P(A) = P(A and B) + P(A and B') = P(A) P(B) + P(A and B'). So P(A and B') = P(A) - P(A) P(B) = P(A) P(B') since P(B') = 1 - P(B). P(B) = P(A and B) + P(A' and B) = P(A) P(B) + P(A' and B). So P(A' and B) = P(B) - P(A) P(B) = P(B) P(A') since P(A') = 1 - P(A).

P(A' and B') = P((A or B)') = 1 - P(A or B) = 1 - (P(A) +P(B) - P(A)P(B)) = 1 - P(B) - (P(A)(1-P(B)) = P(B') - P(A) P(B') = P(B') (1-P(A)) = P(A') P(B')