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?
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.
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