If A and B are independent show that A and B are also indepe

If A and B are independent show that A and B are also independent.

Solution

If the two sets are independent then

P( A and B) = P(A) . P(B)

P(A\') = (1 - P(A) )

P(B\') = (1 - P(B) )

(1-P(A))(1-P(B)) = 1 - P(A) - P(B) - P(A int B)

=> 1 - P(A) - P(B) - P(A)P(B)

=> (1-P(A))(1-P(B))

=> P(A\')P(B\')

Hence the A\' and B\' are also independent