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 $\displaystyle X_n$ denote the number of black balls in urn 1 after the $\displaystyle n$th interchange.

(a) Give the transition probabilities of the Markov chain $\displaystyle X_n$, $\displaystyle n \geq 0$.

(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

$\displaystyle P_{i, i+1} = \frac{(m - i)^2}{m^2}$

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

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

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

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