If A and B are independent event, show that the following pairs of events are also independent. (a) A and B' (b) A'and B'

Question

where A' is the complement of A and B' is the complement of B

Answer 1

P(A∩B) = P(A) P(B)  by the definition of independence

= P(A) (1-P(B'))   since P(B) = 1- P(B')

= P(A) - P(A) P(B')  

So,

 

(1)   P(A) P(B') = P(A) - P(A∩B) 

 

Since A∩B'  =  A - A∩B and A∩B ⊂ A, 

 

(2)   P(A∩B') = P(A) - P(A∩B)

 

From (1) and (2), P(A∩B') = P(A) P(B'), so A and B' are independent

 

This argument shows that if two events are independent, then each event is independent of the complement of the other.  Therefore A' and B' are also independent events.