I know that event with zero probability is independent of any other (because $P(A\cap B)=P(A)\cdot P(B)$). I've got mixed feelings about this. Consider disjoint events $A$ and $B$, where $P(A)=0$. Isn't it weird to say that $A$ and $B$ are independent? It seems to me that it's just as natural to call them "dependent" (since they are disjoint) as to call them "independent" (since $P(A|B)=P(A)$, assuming that $P(B)>0$). I guess my questions are:
- Is there any intuition/motivation on this?
- In other words why do we do it that way? Why not just stick to a definition by conditional probability instead of the product one and leave zero probability events out of the dependence business?