How do I prove that p (A∩B) = p(A) p(B)?

Answer 1

You can’t prove this because this is only true if A & B is independent .

If A & B are independent then :

Pr(A|B) = Pr(A) (since A does not depend on B)

so Pr (A|B) = Pr ( A ∩ B ) / Pr(B) = Pr(A)

multiply both sides of the equation by Pr(B) gives

Pr ( A ∩ B ) = Pr(A) × Pr(B) .