This is surely a simple problem, but there are so many probability questions with dice, urns, etc. that I have not been able to find an answer to this specific problem. Say you have a pair of $n$-sided dice, each individually fair, but the pair is correlated $\rho$. What is the probability, given $\rho$, of throwing the dice one time and rolling the same number on both dice?
The equivalent urn problem is: What is the probability of drawing the same color marble from two different urns, each with $n$ marbles of (the same) $n$ colors, with color somehow correlated $\rho$ between urns?