Show that if A and B are independent then A and Bc are indep

Show that if A and B are independent, then A and B^c are independent.

Solution

Since A and B are independent so

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

P(B|A) = P(A and B) / P(A) = P(B)

Now by conditional probability

P(B\' and A) = P(A) P(B\' |A)

By complement rule

P(B\' and A) = P(A) P(B\' |A) = P(A) [1 - P(B |A)]

Now

P(B\' and A) = P(A) P(B\' |A) = P(A) [1 - P(B and A) / P(A) ]= P(A) [1 - P(B) ] = P(A) P(B\')

Since P(B\' and A) = P(A) P(B\') so A and B\' are independent.