In the book "Introduction to Probability" by J. Charles M. Grinstead and Laurie Snell independent events are introduced in the following way: "It often happens that the knowledge that a certain event $E$ has occurred has no effect on the probability that some other event $F$ has occurred, that is, that $P (F |E) =P (F )$". This is then taken as the definition (notice that the setting in which this is done is that of discrete probabilities - if this makes any difference).
But I can't come to terms with this definition, because of the following two reasons:
a) If I have a die then the event of getting a face with one of the numbers $1,2$, if rolling once, seems intuitively to be independent of getting a face with one of the numbers $5,6$. But since these two events are disjoint, but neither is the empty space, by the above definition, they should be dependent, which seems very counter-intuitive.
b) If we chose to somehow "interpret visually" this definition, then that would mean that the sum of the probabilities of all elements in $F$ weighted with the weight $1$ is equal to the sum of all probabilities in $F\cap E$ weighted with $\frac{1}{P(E)}$, because $$\sum_{\omega\in F} \omega=P (F )=P (F |E) =\frac{P(F\cap E)}{P(E)}=\sum_{\omega\in F\cap E} \omega \cdot \frac{1}{P(E)}.$$
Or - if I use the equivalent definition of independence, $P(E\cap F)=P(E) P(F)$ - then the same interpretation of this definition would say that the sum of all elements in $F$ multiplied the sum of all elements in $E$, both weighted with the weight $1$, is again equal to the sum of all probabilities in $F\cap E$ weighted with $\frac{1}{P(E)}$.
But both these interpretations seem artificial - out of this I can't deduce, why this definition is calling $E,F$ "independent". Could you please solve this "paradox" for me ?
If I were to give a definition of independent events, I would say that $E,F$ are independent (which, for me, would mean "they don't have anything to do with each other"), if $E\cap F =\emptyset$. Why would this be a bad definition ?