We consider an urn containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously $n$ balls from the urn. We number from 1 to $pA$ each of the white balls and, for any natural integer $i \in \llbracket 1, pA \rrbracket$, we define $$Y_i = \begin{cases} 1 & \text{if the ball numbered } i \text{ was drawn,} \\ 0 & \text{otherwise.} \end{cases}$$ Let $Y = \sum_{i=1}^{pA} Y_i$ be the number of white balls drawn. Deduce the value of the variance of $Y$. Compare it to that of $Z$ (where $Z \sim \mathcal{B}(n,p)$).
We consider an urn containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously $n$ balls from the urn. We number from 1 to $pA$ each of the white balls and, for any natural integer $i \in \llbracket 1, pA \rrbracket$, we define
$$Y_i = \begin{cases} 1 & \text{if the ball numbered } i \text{ was drawn,} \\ 0 & \text{otherwise.} \end{cases}$$
Let $Y = \sum_{i=1}^{pA} Y_i$ be the number of white balls drawn. Deduce the value of the variance of $Y$. Compare it to that of $Z$ (where $Z \sim \mathcal{B}(n,p)$).