You have 30 balls 20 black and 10 white let $X_{n}$ be the current number of white balls in urn 1 and each urn holds 15 balls, find
$P_{r}$ ($X_{n+1}$)=k |$X_{n}$=j).
You are taking a ball from urn 1 and urn 2 and placing the ball from urn 1 into 2 and the ball from urn 2 and putting it in urn 1. I should have clarified that earlier.
So I have this representation Urn 1: \begin{equation} \frac{X_{n}\:\:\:15-X_{n}}{15}; \end{equation} with $15-X_{n}$ representing black balls in urn 1
Urn 2: \begin{equation} \frac{10-X_{n}\:\:\:15-(10-X_{n})}{15}; \end{equation}
with $\frac{15-(10-X_{n})}{15}$ being the black balls and $\frac{10-X_{n}}{15}$ being the white balls in urn 2.
So would my answer be $P_{r}$ ($X_{n+1}$)=k |$X_{n}$=j)=$ \frac{15-(10-X_{n})}{15}; $
I realize that my variables might not match up at this point; but I think I'm on the right track but I am just missing something.