Non-independent two consecutive draws from two urns

[hwangii]

Suppose there are two urns: in urn A, there are r red balls and w white balls. In urn B, there are b black balls.

Suppose we do the following experiment: draw k balls from urn A. Among those k balls, put only the red balls in urn B, and draw n balls from urn B. Then the number of red balls from the second draw is a random variable.

Call the random variable $\tilde{y}$. Then
\begin{align*}
Pr(\tilde{y}=y)=\sum\limits_{x=\max\{y,k-w\}}^{\min\{r,k\}}\frac{{r\choose x}{w\choose k-x}}{{r+w \choose k}}\frac{{x\choose y}{b\choose l-y}}{{x+b\choose l}}
\end{align*}

Does anyone know what the mean and the variance of this random variable are? If you do not know the exact form, what about the asymptotic mean and variance when r, w and b go to infinty with the ratio amongst them constant.

Thanks a lot!

 

[mfb]

For r,w,b -> infinity with constant ratio, with constant n, the probability goes to 0 as we have many black balls and nearly no red balls in the second step.

For r,w -> infinity with constant n,b, the first drawing becomes a binomial distribution as function of k.

In general: Expand your (n choose k) as factorials, simplify, approximate them with the Stirling formula, simplify, and see what you get.

For r,w,b,n -> infinity, gaussian distributions are good.