SOLUTION: Suppose that A and B are independent. Show that A' is independent of B' .

Let U be the universal set, to which A and B are the subsets.


Then  (A' ∩ B')  are those elements of the universal set U

that belong neither A nor B.  In other words,  (A' ∩ B') = U \ (A U B).


    Therefore,  P(A' ∩ B') = 1 - P(A U B).


From the other side,  P(A U B) = P(A) + P(B) - P(A ∩ B),  according to the basic formula
of the elementary probability theory.


So we have

    P(A' ∩ B') = 1 - P(A U B) = 1 - P(A) - P(B) + P(A ∩ B) = (1-P(A))*(1-P(B)) = P(A')*P(B').


Thus we proved that  P(A' ∩ B') = P(A')*(P(B').


It means that events A' and B' are independent.