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 denote the number of black balls in urn 1 after the th interchange.
(a) Give the transition probabilities of the Markov chain , .
(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.
Attempted solution:
Transition probabilities
I think those are the transition probabilities. Where is the current number of black balls.
I also know that in order to be time reversible I need the stationary distribution to satisfy: , where is the stationary distribution.
Any assistance as to how I can guess the correct form of the solution based on intuition regarding this scenario would be helpful, thanks in advance.