Question

Prove: If A and B are independent, then so are A and B

Answer 1

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A and B are independent. Then :

P(Ac Intersection Bc) = 1-P(A U B) = 1- (P(A)+P(B)-P(A Intersection B).......1
Since A and B are independent , P(A intersection B) = P(A)*P(B)
Therefore, 1 reduces to:
P(Ac Intersection Bc)
= 1-P(A U B)
= 1- (P(A)+P(B)-P(A)*P(B)).......1
= (1-P(A))*(1-P(B))
= P(Ac)*P(Bc)

So, P(Ac intersection Bc) = P(Ac)*P(Bc)

<Hence, proved>