Please explain each step and write clear Let A and B be two

Please explain each step and write clear.

Let A and B be two independent events. Do you think that A\' and B are also independent? Prove it.

Solution

A and B are independent events

So P(A n B) = P(A).P(B)

Now, P(A\')P(B) = (1-P(A))P(B) = P(B) - P(A).P(B)

But P(A).P(B) = P(A n B)

So P(A\')P(B) = P(B) - P(A n B)

We also know that A\' n B = B - (A n B)

So P(A\')P(B) = P(A\' n B)

So A\' and B are also independent events