Question

Let A and B be events with P(A) = .4, P(B) = .4 , and P(A U B) = .7 . Are A and B independent? Why?

Answer 1

Two events A and B are independent if p ( A B ) = p ( A ) * p ( B )

We know the formula of union of two events :

p ( A B ) = p ( A ) + p ( B ) - p ( A B )

0.7 = 0.4 + 0.4 - p ( A B )

0.7 = 0.8 - p ( A B )

0.7 - 0.8 = - p ( A B )

-0.1 = - p ( A B )

p ( A B ) = 0.1

p ( A ) * p( B ) = 0.4 * 0.4 = 0.16

So here p ( A B ) p ( A ) * p ( B)

So two events are not independent.