Expected number of points in a subsurface of a rectangle There is a rectangle $S$, and $n$ points are uniformly distributed inside it. If we select an area $A$ inside the rectangle, what is the expected number of points inside the $A$?
I think it seems to be $\dfrac{area(A)}{area(S)}$, but some one told me the distribution of points is discrete but the way I calculate is continuous.

You've almost got it:
How about letting the number $X$ of points in $A$ have
the discrete distribution $\mathsf{Binom}\left(n=n_S. 
p=\frac{\mathrm{area}(A)}{\mathrm{area}(S)}\right),$
where $n_S$ is the number of points randomly selected within $S?$
As an obvious example, let $A$ be the "left" half of $S$
and randomly select $n_S = 50$ points in $S.$
Then $X \sim \mathsf{Binom}(50, 0.5)$ with mean $E(X) = 25.$
