## Time reversible Markov Chain

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 $X_n$ denote the number of black balls in urn 1 after the $n$th interchange.
(a) Give the transition probabilities of the Markov chain $X_n$, $n \geq 0$.
(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.

Attempted solution:
Transition probabilities
$P_{i, i+1} = \frac{(m - i)^2}{m^2}$

$P_{i,i} =\frac{2(m - i)(i)}{m^2}$

$P_{i, i - 1} = \frac{i^2}{m^2}$

I think those are the transition probabilities. Where $i$ is the current number of black balls.

I also know that in order to be time reversible I need the stationary distribution to satisfy: $\pi_iP_{ij} = \pi_jP_{ji}$, where $\pi$ 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.