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In the Bernoulli-Laplace diffusion model, two urns each contain $n\ge 1$ 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 $\{0,\dots ,n\}$, withThe upward event exchanges a blue ball from the first urn with a red ball from the second; the downward event reverses the colors. HenceStarting from $R_0=n$, the mean is $n/2+(n/2)(1-2/n{)}^k$, with the initial value understood separately if the geometric factor is zero. For $n\ge 2$, this mean tends to $n/2$. For $n=1$, however, the only two configurations alternate and the expected imbalance does not decay.

The Stationary distribution is the hypergeometric distributionIt counts uniformly allocated red/blue configurations. The ratio $\pi (r+1)/\pi (r)=[(n-r)/(r+1){]}^2$ verifies Detailed balance with the displayed transitions. The chain is irreducible; for $n\ge 2$ an interior holding Probability also makes it aperiodic, while the $n=1$ chain has period two. This model separates convergence of an average imbalance from the full probability law and its periodicity.

Conditional on $R_k=r$, urn 1 has $r$ red and $n-r$ blue balls, while urn 2 has $n-r$ red and $r$ blue balls. The independent selections therefore giveEvery other conditional probability is zero. The out-of-range terms at $r=0$ or $r=n$ already have zero probability. These are the transition probabilities of the Bernoulli-Laplace diffusion model.

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.