Derive a Combinatorial Identity

Question

An urn exists with $x$ red balls and $y$ green balls. Draw $n$ balls without replacement.

Derive a combinatorial identity for nonnegative integers $n$, $x$, and $y$, satisfying $1 \leq n \leq x + y$ from consideration of the following:

$$na = \sum_{k=0}^n k {n \choose k}a^k (1-a)^{n-k}$$

Attempt: I know that one side of the identity is $$n{x\over x+y}$$

Edit: The question is not to derive the na = ... identity above. It is to derive a new identity, using that identity as a guide, for drawing balls from the urn without replacement.

Answer 1

\begin{aligned} \sum^n_{k=0}k\binom{n}{k}a^k(1-a)^{n-k}&=\sum^n_{k=1}\frac{n(n-1)!}{(k-1)!(n-k)!}a^k(1-a)^{n-k}\\ &=na\sum^n_{k=1}\binom{n-1}{k-1}a^{k-1}(1-a)^{(n-1)-(k-1)}\\ &=na\sum^n_{k=0}\binom{n-1}{k}a^k(1-a)^{(n-1)-k}\\ &=na(a+1-a)^{n-1} \end{aligned}

Answer 2

Oliver's answer is great, but just in case you like derivatives like me,

\begin{align} \frac{\partial}{\partial y}(x+y)^n = \sum_{k=0}^n { n \choose k} x^{n-k} \frac{\partial}{\partial y}y^k\newline n(x+y)^{n-1}=\sum_{k=0}^n { n \choose k} k x^{n-k} y^{k-1} \end{align}

Taking $x = (1-a)$ and $y = a$, $x+y = 1$,

\begin{align} na=\sum_{k=0}^n { n \choose k} k (1-a)^{n-k} a^{k} \end{align}

Answer 3

The expression provided as reference is the expected number of balls, having prob. $a$ of being extracted with replacement from an urn containing a total of $n$ balls.

$n x/(x+y)$ is the expected number of balls "type x" in $n$ balls extracted without replacement from $x+y$ total balls: a sample mean.

You have $\binom{x}{k}$ ways to extract a $k$-subset from the set of $x$ balls (labelled) and $\binom{y}{n-k}$ to extract $n-k$ from the $y$. In total $$ \sum\limits_{\left( {0\, \le } \right)k\,\left( { \le \,n} \right)} {\left( \matrix{ x \cr k \cr} \right)\left( \matrix{ y \cr n - k \cr} \right)} = \left( \matrix{ x + y \cr n \cr} \right) $$ ways.

Then $$ n{x \over {x + y}} = {1 \over {\left( \matrix{ x + y \cr n \cr} \right)}}\sum\limits_{\left( {0\, \le } \right)k\,\left( { \le \,n} \right)} {k\left( \matrix{ x \cr k \cr} \right)\left( \matrix{ y \cr n - k \cr} \right)} $$