Expection and Variance:
 If Y is a random variable with a hypergeometric distribution,then,
$$E(Y) = frac{{nr}}{N};;;{rm{and}};;;V(Y) = nleft( {frac{r}{N}} right)left( {frac{{N - r}}{N}} right)left( {frac{{N - n}}{{N - 1}}} right).$$
 You can use the following Mathematica command to obtain these results
    Expectation[x, x [Distributed] HypergeometricDistribution[n, r, N]]
 or Mean[HypergeometricDistribution[n, r, N]]
    Variance[HypergeometricDistribution[n, r, N]]
Property:
  For a fixed fraction $p = frac{r}{N}$, the hypergeometric probability function converges to the binomial probability function as $N$ becomes large and $n$ is relatively small.
$$mathop {lim }limits_{N to infty } frac{{left( {begin{array}{*{20}{c}} r\ y end{array}} right)left( {begin{array}{*{20}{c}} {N - r}\ {N - y} end{array}} right)}}{{left( {begin{array}{*{20}{c}} N\ n end{array}} right)}} = left( {begin{array}{*{20}{c}} n\ y end{array}} right){p^y}{left( {1 - p} right)^{n - y}}.$$