This is probably a very stupid question but I can't wrap my head around it.
$$ P(B \cap A) = P(A \cap B) = P(B \mid A)\cdot P(A) + P(A \mid B)\cdot P(B) $$
Can someone explain intuitively why the above isn't true? What I am essentially saying here is that A and B both occurred is same as saying A occurred and then B occurred,following A OR B occurred and then A occurred, following B. So probability of A AND B happening = (Probability that A happened AND then probability that given A, B happened) OR (Probability that B happened AND then probability that given B, A happened)
Where am I going wrong?