TITLE: Condition of Statistical Independence
QUESTION [1 upvotes]: 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.

REPLY [1 votes]: $\Pr(A|B)\Pr(B)=\Pr(A \cap B)$ 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  $\Pr(A|B)\Pr(B)=\Pr(B|A)\Pr(A)$.
{"set_name": "stack_exchange", "score": 1, "question_id": 113593}