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!
d4
. In one case, the pairs of results (1,2), (2,1), (3,4), and (4,3) have equal chances. The correlation coefficient is $\rho = 3/5$ and there is zero chance the two results are equal. In another case the outcomes (1,1), (2,2), (3,3), (4,4) each have a chance of $1/5$ and the outcomes (1,4), (2,3), (3,2), and (4,1) each have a $1/20$ chance. Again $\rho = 3/5$ but now there is an 80% chance the two dice show the same value. $\endgroup$