A total of m white and m black balls are distributed among two urns, with each urn containing m balls. At each stage, a ball is randomly selected from each urn and the two selected balls are interchanged. Let Xn denote the number of black balls in urn 1 after the nth interchange.

(a) Give the transition probabilities of the Markov chain Xn, n 0.

(b) Without any computations, what do you think are the limiting probabilities of this chain?

(c) Find the limiting probabilities and show that the stationary chain is time reversible.