Variance of 7 drawn balls out of an Urn. We have an urn with 40 balls out of which 10 are characterized as "1" and the remaining 30 as "0". We pull 7 balls out of the urn without placing them back. 
$Z_{i}$ denotes the number of the i-th drawn ball.
Find $\mathrm{Cov}(Z_1,Z_2)$ and $\mathrm{Var}(Z_2)$
My thought: They must be somehow hypergeometrically distributed?
This shall be my last elementary probability question before my exam tomorrow..
 A: The expected value of any $Z_k$ is $\frac14$. The variance of any $Z_k$ is
$$
\frac1{40}\left[10\left(\frac34\right)^2+30\left(-\frac14\right)^2\right]=\frac3{16}
$$
The covariance of any two $Z_j$ and $Z_k$ is
$$
\frac1{\binom{40}{2}}\left[\vphantom{\left(\frac34\right)^2}\right.\overbrace{\binom{10}{2}\left(\frac34\right)^2}^{\text{both }1}+\overbrace{\binom{10}{1}\binom{30}{1}\frac34\cdot\left(-\frac14\right)}^{\text{a $1$ and a $0$}}+\overbrace{\binom{30}{2}\left(-\frac14\right)^2}^{\text{both }0}\left.\vphantom{\left(\frac34\right)^2}\right]=-\frac1{208}
$$
A: Yes, they are hypergeometrically distributed. 
Do you know what the specific distribution of a subset of a hypergeometric distribution is?
Does it matter in what order the balls are drawn?
When you know the answers to these questions the rest follows mechanically from the definitions of covariance and variance.
A: Let each ball take the value of 0 or 1. If it is so, find the probability of the Z_i ball being a "0" or "1" and the below table illustrates the use of E(Z), Var(Z) and COV(Z1,Z2).
Thanks
Satish

