Suppose we have two boxes containing $M$ balls each. At each time increment, one ball is uniformly selected from each box, and then placed in the other one. If at time $0$ the first box contains $M$ white balls and the second one contains $M$ black balls, define $X_{n}$ to be the number of black balls inside the first box at time $n$.
I have already computed all the transition probabilities, but I'm now looking to determine $E(X_{n+1}|X_{n})$ for $n=0,1, \dots$ and then to show that $\lim_{n \to \infty} E(X_{n}) = M/2$.
To do this, I've reasoned that for each time $n$, $X_{n}$ can at most be at state $M$. So I tried working out $$E(X_{n+1}|X_{n}=j)= \sum_{k=1}^{\max(n+1,M)} kP(X_{n+1} = k| X_{n} = j) $$
where $j \in \{0, \dots, \max(n,M)\}$. I don't think I need to do double summation since there's a lot of zero terms). That is, I can restrict attention to $j \in \{k-1, k, k+1\}$, but the problem is still messy. And things are complicated when $k \in \{0,M\}$.
There have been some difficulties working this out, so I'm here to ask if there's something more simple I should be doing? It seems like this path would get needlessly messy.
Edit: I could post the transition probabilities, if anyone is interested.
Let's find this using identifier RVs. Let $I_{1\to2}$ be equal to 1 if the ball going from bin 1 to bin 2 is black at time $n+1$ and 0 otherwise. Similarly define $I_{2\to1}$. We now have \begin{align*} E[X_{n+1}|X_n]&=X_n - E[I_{1\to2}|X_n] + E[I_{2\to1}|X_n]\\ &=X_n-P[I_{1\to2}=1|X_n]+ P[I_{2\to1}=1|X_n]\\ &=X_n-\frac{X_n}{M}+\frac{M-X_n}{M}\\ &=X_n\left(1-\frac2M\right)+1. \end{align*} So $$ E[X_{n+1}]=E[E[X_{n+1}|X_n]]=E[X_n]\left(1-\frac2M\right)+1. $$
By finding the limit of the recurrence $$a_{n+1}=a_n\left(1-\frac2M\right)+1,\quad a_0=0,$$ you should be able to prove $E[X_n]\to M/2$.