Correlation coefficient between two binominals My problem is the following: Roll a dice 30 times, X is the number of 1's and Y is the number of 6's. Find the correlation coefficient $R(X,Y)=\frac{\operatorname{Cov}(X,Y)}{\sigma_x \sigma_y}$. 
The covariance is $\operatorname{Cov}(X,Y)=E(XY)-E(X)E(Y)=0-5*5=-25$ and the product of the ${\sigma}$-s is  $30\cdot\frac{1}{6}\cdot\frac{5}{6}=\frac{25}{6}$, that gives $R=-6$ which is a bad solution for $R$. 
Where did I mistake? Can anybody fix it? Thank you for your answers
 A: HINT: $E[XY] \neq 0$
Let $X_i$ be the number of $1$s rolled by the $i$th die; so $X_i = 1$ or $0$, and $X = \sum_{i=1}^{30} X_i$.  Similarly let $Y_i$ be the number of $6$s rolled by the $i$th die.
As you already know, $E[X] = E[\sum X_i] = \sum E[X_i] = 30 \times {1 \over 6} = 5$ and same for $E[Y]$.
Now what is $E[XY]$?  As you pointed out in the comments, it is true that $X_i Y_i = 0$ since one die cannot show both a $1$ and a $6$.  However, $XY \neq \sum_{i=1}^{30} X_i Y_i$!!  And they are certainly dependent (e.g. $X=29 \implies Y \le 1$).
To calculate this, just expand: 
$$XY = (\sum X_i) (\sum Y_i) = \sum_i X_i Y_i + \sum_{i \neq j} X_i Y_j$$
This has $30 \times 30 = 900$ terms in it, but you can calculate the expectation of every term easily.  Can you finish from here?
A: If you threw one die, then using your method you would get  $\text{Cov}(X,Y)= 0-\frac16 \times \frac16=-\frac1{36}$  and since the variances are each ${\frac{5}{36}}$ the standard deviations are each $\sqrt{\frac{5}{36}}$, and you would get a correlation coefficient of $\frac{-1/36}{5/36}=-\frac15$
For the sums over $30$ independent dice throws, you can immediately say this $-\frac15$ is again the correlation coefficient, but if you prefer a little more explanation, then for the sums over $30$ independent dice throws you multiply the covariance by $30$ to give $\text{Cov}(X,Y)= -\frac{30}{36}=-\frac{5}{6}$ and each of the variances also by $30$ to give ${\frac{150}{36}}={\frac{25}{6}}$ so each standard deviation is $\sqrt{\frac{25}{6}}$, giving a correlation coefficient of $\frac{-5/6}{25/6}=-\frac15$ 
