Prove that, complement of two independent events A and B also independents.

 Q. Prove that, complement of two independent events A and B also independents. 

Proof:

    We know if  A and B is independents then P( A∩B) = P(A) . P(B)
    Also, know that P(A') = 1 - P(A) and P(B') = 1 - P(B)

    Now, P(A'∩B') = P((A∪B)')                   [Using De Morgan's]
                            = 1 - P(A∪B) 
                            = 1 - P(A) - P(B) - P(A∩B)
                            = [1 - P(A)] - P(B) - P(A) . P(B)
                            = [1 - P(A)] - P(B) [1 - P(A)]
                            = [1 - P(A)] . [1 - P(B)]
                            = P(A') . P(B')

 Therefore, P(A'∩B') = P(A') . P(B'), So, we can say that, complement of two independent events A and B also independents. ☺