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I have this question:

P(A | B)P(B) = P(B | A)P(A) if A and B are independent events.

My reasoning: If A & B are independent, P(A|B) = P(A) and P(B|A) = P(B) and the above expression reduces to P(A)*P(B) = P(B)*P(A), which proves that the statement is true.

However the answer indicates that it is not true. Any mistakes in my argument?

Thanks.

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    This holds true whenever A and B are independent or not. Hint: recall the definition of P(A|B).2012-02-26

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$\Pr(A|B)\Pr(B)=\Pr(A \cap B)$ whether A and B are independent or not.

Similarly $\Pr(B|A)\Pr(A)=\Pr(B \cap A)$.

And $A \cap B = B \cap A$ since intersection is a commutative function.

So $\Pr(A|B)\Pr(B)=\Pr(B|A)\Pr(A)$.