There are 3 red marbles in jar A and 2 blue marbles and 1 red marble in jar B. Every time, one simultaneously draws a marble at random from jar A and jar B and place the marbles drawn into the opposite jars. Letting $X_n$ denote the number of blue marbles in jar A after $n$ times, find the PTM for this process.
I filled out what I could just by counting outcomes. Please let me know if this is correct/if there's a quicker way.
$X_n$ can be either $0, 1, 2$. The columns go from 0 to 2 rightward and the rows go from 0 to 2 downwards.
$P = \begin {bmatrix} \frac{1}{3} & \frac{2}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3}\\ 0 & 1 & 0 \end {bmatrix}$
And if this PTM is correct, is the probability of A having no blue marbles after 2 steps $\frac{1}{3}$?