Answer to Question #166479 in Statistics and Probability for phemelo mongae

Question

Show that if events A and B are independent, then the following pairs of events are also independent. (a) A and B¯ (b) A¯ and 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.