SOLUTION: Prove that if two events A and B are independent, then so are their complements.

Question 1027777: Prove that if two events A and B are independent, then so are their complements.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
If A, B are independent events, then by definition, P(A∩B) = P(A)P(B).
We have to show that P(A'∩B') = P(A')P(B').
P(A'∩B') = P((A∪B)') = 1 - P(A∪B) = 1-P(A) - P(B) -P(A∩B)
=1 - P(A) - P(B) -P(A)P(B) = (1 - P(A))(1 - P(B)) = P(A')P(B')

Thus, P(A'∩B') = P(A')P(B'), and the statement is proved.
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