I am trying to solve this exercise:
At time $0$, a box contains one white ball and one black ball. At each time n = 1,2,3,4..., one ball is randomly choosen. After that, the ball is re-inserted on the box and a new ball with the same color is inserted in the box. Hece, at n, the box is going to have (n+2) balls. Define B_n as the number of black balls chosen untill time n. Show that this process is a Markov Chain and present the transition matrix.
I have a partial solution but I think my proof is wrong. I know the definition of a markov chain and I have read similar questions like this one on Stack Exchange. However, I was not able to make a decent proof.
That'st my try on the transition matrix for the problem above (probably wrong):
$$ \begin{bmatrix} 1/2 & 1/2 & 0 & 0 &0 &0 &\cdots \\ 0& 1/3 & 1/3 & 1/3 &0 &0 &\cdots \\ 0 & 0 & 1/4 & 1/4 & 1/4 &1/4 &\cdots \\ \vdots & &&\ddots \end{bmatrix} $$
Thanks in advance.