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?
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.