(Bernoulli–Laplace model of diffusion). Consider two urns each of which contains m balls; b of these 2m balls are black, and the remaining 2m[1]b are white.

We say that the system is in state i if the first urn contains i black balls and m [1] i white balls while the second contains b [1] i black balls and m [1] b C i white balls.

Each trial consists of choosing a ball at random from each urn and exchanging the two. Let Xn be the state of the system after n exchanges have been made. Xn is a Markov chain.

(a) Compute its transition probability.

(b) Verify that the stationary distribution is given by .i/ D b i ! 2m [1] b m [1] i !, 2m m !

(c) Can you give a simple intuitive explanation why the formula in

(b) gives the right answer?