Probability in permutation of balls

Question

There are $n$ balls, of which $r$ balls are red and $(n-r)$ balls are blue.

If we select $d$ balls at random (without replacement), what is the probability we select $rd/n$ red balls? In the asymptotic case, can we show concentration around $rd/n$?

Answer 1

The expectation for the amount red balls in $n$ can be found:

$$\begin{align} E(r) &= \dfrac{n}{2} \\ \\ E(d) &= \dfrac{n}{2} \\ \\ E(d=r) &= \frac{\frac{n}{2}}{2} = \frac{n}{4} \\ \end{align}$$

Then $$E\left(\dfrac{rd}{n}\right) = \dfrac{E(r) * E(d)}{n} = \dfrac{\frac{n}{2}*\frac{n}{2}}{n} = \dfrac{\frac{n^2}{4}}{n} = \frac{n}{4}$$