Probability of same outcome for a pair of dice/urn draws with correlation $\rho$ 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?
Thanks!

Without additional constraints, this question cannot be answered, because there are many different configurations that lead to the same correlation coefficient.
You can think of this problem in terms of a table of probability distribution across all possible pairs of values, e.g. (with many values blank out of laziness)





a=1
a=2
a=3
a=4




b=1
p(1,1)
p(2,1)




b=2
p(1,2)
p(2,2)




b=3


p(3,3)



b=4



p(4,4)




One way to obtain a strong correlation is to have high probabilities on the diagonal - where both die have the same value. However, another would be to have dice b always having a value one greater than dice a (except when dice a is already at its maximum).
What you can do, however, is specify different probability matrices, and see what correlations they would correspond to.
Update
It's interesting to think about the kinds of constraint you could impose here that would lead to unique solutions. One approach would be to constrain the model so that dice B will either have the same value as A, with probability $m$, or be randomly sampled with equal probabilities from the other $n - 1$ possible values (or from all $n$ possible values).
