Bernoulli-Laplace diffusion model ($P(r,r+1)=(n-r{)}^2/n^2,P(r,r-1)=r^2/n^2$)

Two urns each contain $n$ balls, with $n$ red and $n$ blue balls in total. Independently choose one ball uniformly from each urn and exchange them. The red count $R_k$ in the first urn is a Markov chain on $\{0,\dots ,n\}$. Its upward and downward probabilities are displayed above, and the probability of remaining at $r$ is $2r(n-r)/n^2$. Thus $\mathbb{E}[R_{k+1}\mid R_k=r]=1+(1-2/n)r$. Starting from $R_0=n$, linearity of expectation gives $\mathbb{E}[R_k]=n/2+(n/2)(1-2/n{)}^k$, with the initial value separately understood when $n=2,k=0$. Random exchanges drive the expected imbalance toward zero.