We have an urn with $6$ red balls and $4$ green balls. We draw balls from the urn one by one without replacement, noting the order of the colors, until the urn is empty. Let $X$ be the number of red balls in the first five draws, and $Y$ the number of red balls in the last five draws. Compute the covariance $Cov(X,Y)$.
I am struggling to "split" up the events of the first and last five balls in order to get an appropriate probability. I've tried creating a joint probability table where the rows are $X=1,2,3,4,5$ and columns restart for $Y=1,2,3,4,5$ However, I feel this is the wrong approach.
$X$ and $Y$ are identically distributed hypergeometric distributions. Thus \begin{align*} \text{Var}(X+Y)&= \text{Var}(X)+\text{Var}(Y) +2\text{Cov}(X,Y)\\ 0&=2\text{Var}(X)+2\text{Cov}(X,Y)\\ \text{Cov}(X,Y) &= -\text{Var}(X)\\ \text{Cov}(X,Y) &=-\frac{2}{3}. \end{align*}