We place $N$ balls in $K$ baskets (in whatever way) and perform the following discrete-time process: at each time $t \in \Bbb N$ starting at $t = 0$ we choose one of the balls so that each ball is chosen with probability $\frac 1N$ and place it in one of the baskets, so that each basket is chosen with probability $\frac 1K$. Denote by $X_n$ the number of balls in the first basket at $t = n$. Denote by $ F_n$ the natural filtration generated by $X_n$. Compute $E[X_{n+1}|F_n]$.
I figured out that $$P(X_{n+1}=X_n + 1)= \frac{(K-1)X_n}{NK} $$ $$P(X_{n+1}=X_n - 1)= \frac{N-X_n}{NK} $$ $$\text{and } P(X_{n+1}=X_n )= 1-P(X_{n+1}=X_n + 1) - P(X_{n+1}=X_n - 1) $$ so I guess that $X_{n+1}=X_n + Y$ where $Y$ is a random variable taking values in $\{0,-1,1\}$ with the probabilities above. Hence, I'd have that $E[X_{n+1}|F_n] = X_n + E[Y|F_n]$ but then I don't know what to do... Here, $E[X_{n+1}|F_n]$ is a random variable and I cannot use a formula for $E[X_{n+1}|X_n=x]$ which is a real number. What can I do ?
Recall that probabilities are numbers, not random variables! That is, the right-hand side should not include $X_n$. Instead, what you have computed is: $$P(X_{n+1}=X_n + 1 \, | \, F_n)= \frac{(K-1)X_n}{NK} $$ $$P(X_{n+1}=X_n - 1 \, | \, F_n)= \frac{N-X_n}{NK} $$
Thus, \begin{align} \mathbb{E}[X_{n+1} \, | \, F_n] &= (X_n+1) P(X_{n+1}=X_n + 1 \, | \, F_n) + (X_n-1) P(X_{n+1}=X_n - 1 \, | \, F_n) + X_n (1- P(X_{n+1}=X_n - 1 \, | \, F_n) - P(X_{n+1}=X_n + 1 \, | \, F_n)) \\ &= P(X_{n+1}=X_n + 1 \, | \, F_n) - P(X_{n+1}=X_n - 1 \, | \, F_n) + X_n \\ &= X_n + \frac{KX_n - N}{NK} \end{align}