(I'll post my own answer to this, but others may be of interest, so post your own if you have one.) (**PS:** In reply to comments posted below: Stackexchange [encourages posting an answer to one's own question](https://math.stackexchange.com/help/self-answer).)

Paul and Tatiana Ehrenfest posed the problem of two urns containing some marbles. At each step in a random process, one marble is chosen randomly, all having the same chance of being chosen. The chosen marble is moved from whichever urn it's in to the other one. The number of marbles in the first urn is the state of a Markov chain. The question is: what is the equilibrium distribution of this Markov chain? The answer is: it is the binomial distribution of the number of successes in $n$ trials with probability $1/2$ of success on each trial, where $n$ is the number of marbles in the two urns. So now the question is: how do you prove that?