Which of the two popular definitions of independent events is more primitive? I know there are two ways to say event $a$ and $b$ are independent:


*

*$P(a)P(b)=P(a\cap b)$

*$P(a\mid b)=P(a)$


and I can derive one from the other with the Bayes Formula $P(a|b)=P(a\cap b)/P(b)$.My question is: Of the two equations above, which is the definition from which the other equation is proven?
 A: After a few days searching, I find it's clearly explained in the wiki pages:

Two events A and B are independent if and only if their joint
  probability equals the product of their probabilities:
$P(A∩B)=P(A)P(B) $
Why this defines independence is made clear by rewriting with
  conditional probabilities: 
Although the derived expressions may seem more intuitive, they are not
  the preferred definition, as the conditional probabilities may be
  undefined if $P(A)$ or $P(B)$ are 0. Furthermore, the preferred
  definition makes clear by symmetry that when $A$ is independent of
  $B$, $B$ is also independent of $A$.

A: If my primitive you mean immediately apparent, and obvious I would say $$P(a)*P(b)=P(ab)$$ but this might be considered subjective, although in practice, that equation is where almost all probability classes start. It is quite intuitively obvious, although to a prodigy Baye's theorem might be "obvious" as well. 
A: I've most often seen $P(A\cap B) = P(A)P(B)$ as the definition of independence, and for independent events $A,B$, $P(A|B)=P(A)$ as a theorem that one proves using the previous definition. So in that sense $P(A\cap B) = P(A)P(B)$ is as "primitive" as it gets.
