Linearity of expectations - Why does it hold intuitively even when the r.v.s are correlated? An experiment - say rolling a die, is performed a large number of times, $n$. Let $X$ and $Y$ be two random variables that summarize this experiment. 
Intuitively(by the law of large numbers), if I observe the values of $X$, over a large number of trials, take their mean, $m_{X}=\frac{1}{n}\sum_{i}{x_{i}}$, and observe the values of $Y$, take their mean $m_{Y}=\frac{1}{n}\sum_{i}{y_{i}}$ and the add the two column means, this is very close to $E(X)+E(Y)$.
If we observe the values of $X+Y$ in a third column, and take their arithmetic mean, $m_{X+Y}$, this will be very close to $E(X+Y)$. 
Therefore, linearity of expectation, that $E(X+Y)=E(X)+E(Y)$ emerges as a simple fact of arithmetic (we're just adding two numbers in different orders).
I know linearity of expectations holds, even when the $X$ and $Y$ are dependent. For example, the binomial and hypergeometric expectation is $E(X)=np$, although in the binomial story, the $Bern(p)$ random variables are i.i.d., but in the hypergeometric story, they are dependent.
If two random variables are correlated, wouldn't that affect the average of their sum, than if they were uncorrelated? Any insight or intuition would be great!

For intuition, suppose the sample space consists of a finite number of equally probable outcomes (this is of course not true for all probability spaces, but many situations can be approximated by something of this form). Then
$$ E(X+Y) = \frac{(x_1+y_1)+(x_2+y_2)+\cdots+(x_n+y_n)}n $$
and
$$ E(X)+E(Y) = \frac{x_1+x_2+\cdots+x_n}n + \frac{y_1+y_2+\cdots+y_n}n $$
which is obviously the same.
