 Solution

At state $r$, urn 1 has $r$ red and $n-r$ blue balls, while urn 2 has $n-r$ red and $r$ blue balls. Conditional on the state, the two choices are independent, each assigning probability $1/n$ to every ball. A red–blue exchange decreases $r$; a blue–red exchange increases it; equal colours leave it unchanged. Thus the conditional probability mass function isThe three numerators sum to $n^2$. At $r=0$ or $n$, the outward transition has probability zero. These are the transitions of the Bernoulli-Laplace diffusion model. The displayed formula is a transition kernel on every state; a literal conditional probability at a particular time requires the conditioning event to have positive probability.