Understanding why conditional probability is defined the way it is. Just moments ago I stumbled upon the definition of conditional probability and I'm experiencing some difficulties in acquiring an intuitive understanding of why it's defined the way it is. 
$$P(E|F)= \dfrac{P(E \cap F)}{P(F)}$$
I'm more familiar with taking into consideration the amount of outcomes of interest and dividing that amount by the amount of outcomes in the sample space. I don't know how to interpret the logic behind conditional probability. I understand that $P(F)$ is the probability that the event which is known to have occured occurs. So I suppose that it can be calulated the way I am familiar with. Presumably the same goes for $P(E \cap F)$ by using the Inclusion–exclusion principle. But why one is divided by the other I cannot fathom. Why is it defined this way? How does one make sense of it.  
 A: What is the probability for two events $E$ and $F$ to both happen? It is the probability of $E$ to happen, times the probability of $F$ to happen, given that $E$ happens. So: $P(E\land F) = P(E)*P(F|E)$ ... from which your formula immediately follows.
Note that as a special case we have that $E$ and $F$ are independent events, meaning that the occurrence or non-occurrence of one event has no bearing on the other, and thus $P(F|E)=P(F)$. And so in that case, $P(E \land F) = P(E)*P(F)$ ... which I assume is a formula you are well familiar with! So we now see that the latter formula is really a special case of the more general $P(E\land F) = P(E)*P(F|E)$.
Something else you can quickly get out of the general formula for two co-occurring events is this: $P(E \land F)$ is of course equal to $P(F \land E)$. Therefore, $P(E)*P(F|E)=P(F)*P(E|F)$, and from this you immediately get Bayes' Law:
$P(E|F) = \frac{P(E)*P(F|E)}{P(F)}$
Cool! The moral is: sometimes you just need to remember one intuitive principle, and a bunch of others can be quikly derived if needed. Indeed, I can't tell you instantly the formula for Bayes's Law if you were to ask me ... But I know how to very quickly derive it when I need it!
