If A is a random variable responsible for calculating the sum of two independent rolls of a die, and B is the result of calculating the value of first roll minus the value second roll, is is true that A and B have a $cov(A,B)\neq 0$? In other words, is it true that they are correlated?
I've come to the conclusion that they must be correlated because they are not independent, that is, the event of A can have an impact on event B, but I remain stuck due to the fact that causation does not necessarily imply correlation.
I know that independence $->$ uncorrelation, but that the opposite isn't true.
Covariance of independent variables is $0$ but covariance of dependent variables is not necessarily non-zero: it might be $0$ (which is exactly what happens in this case), so your conclusion is untrue.
Let $X_1,X_2$ be independent random variables denoting the number rolled on the two fair die respectively. $X_1,X_2$ are identically distributed. $A=X_1+X_2,B=X_1-X_2$.
$\Bbb E[B]=E[X_1]-E[X_2]=0.\\\Bbb E[AB]=E[X_1^2]-E[X_2^2]=0.$
So the covariance $\Bbb E[AB]-\Bbb E[A]\Bbb E[B]$ is $0$.