I have a general doubt on independent events. After studying some basics on probability from A first course in probability, I don't see exactly why the relation for independent events comes from, specially if we derive it from the conditional probability equation and we think on terms of Venn diagrams.
So, if we have 2 events E and F, we say that the conditional probability of E given that F has occurred is $P(E|F)=\frac{P(EF)}{P(F)}$. So, the events E and F are independent if $P(E|F)=P(E)$, or $P(EF) = P(F)P(E)$. Actually, the first way of expressing it is pretty clear: doesn't matter if F occurred or not, the probability of E is the same. But numerically speaking, I fail to see it.
Moving on to Venn diagrams, and trying to portrait this result, the space that E must occupy inside of F needs to be the same as it occupies in proportion to the total sample space; but it means at the same time that they share some space. I can think of some cases where events share space but are independent (easily, taking a ace or a spade from a deck), but I fail to see how to integrate the idea of independence into the diagram.
I fail to understand the meaning of independent events, and how these can be differenced from the cases in which the conditional probability is the same as the event that is conditioned.