There are 2 colorless balls in an urn. In a sequence of events, a ball is randomly chosen and painted in red or black, then we put it back in the urn. We do it again, if the ball is painted, we put replace it whit a different color (if we pick a red ball, we put a black one in the urn). If the ball is not painted, we randomly paint it in red or black.
How can we describe the transition matrix of this process with states $(x,y,z)$, where $x$ is the number of colorless balls, $y$ is the number of red balls and $z$ is the number of black balls.
My first attempt was to study the case when we only have one ball, I get this matrix:
\begin{bmatrix} 0 & 0.5 & 0.5 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}
When I go for the second case, the 2 balls, things get a little confused. I know i can have the following cases: $(2,0,0),(1,1,0),(1,0,1),(0,1,1),(0,2,0),(0,0,2)$.
Attention: I posted a similar question earlier, but now I think it is better written. I've tried to write a similar matrix but i failed. I've even thinked and managed to do an 6x6 matrix, using the possible combinations, but did not maneged to do an 3x3. Any help is appreciated. If i wrote something that isn't clear, please let me know!