How to prove that none of the three events are independent ?

Question

If I have 3 events A,B and C such that all 3 are independent , then how to prove that P(A' ∩ B' ∩ C')  is also independent , for this 

P(A' ∩ B' ∩ C') =1- P(A∪ B ∪C)

                     =1-[ P(A)+P(B)+P(C)- P(A∩B)-P(A∩C)-P(B∩C)+P(A∩B∩C)  ]

                      =1- [ P(A)+P(B)+P(C)-P(A)P(B) -P(A)P(C) -P(B)P(C) +P(A)P(B)P(C) ]

                      =1-[P(A)+P(B)+P(C) -P(A)P(B)-P(A)P(C)-P(A')P(B)P(C) ] 

                      =1 -[P(A)+P(C)+P(A')P(B)-P(A)P(C)-P(A')P(B)P(C) ]

                       =1-[P(A)+P(A')P(C)-P(A')P(B) -P(A')P(B)P(C) ]

                       =P(A') -P(A')P(C) +P(A')P(B) +P(A')P(B)P(C)

                        =P(A')P(C') +P(A')P(B)+P(A')P(B)P(C)

Now after this I am unable to get the actual result which I should be getting for proving that even this (A' ∩ B' ∩ C') is also an independent event , so plz clarify this .

                       

                 

 

Answer 1

if A, B, C are independent events, P(AnBnC)=P(A)P(B)P(C)

=> A',B',C' are also independent events. since P(A)'=1-P(A)

hence P(A'nB'nC')=P(A)'P(B)'P(C)'.