Three fair coins are tossed, and we let $X_1$ denote the number of heads that appear. Those coins that were heads on the first trial (there were $X_1$ of them) we pick up and toss again, and now we let $X_2$ be the total number of tails, including those left from the first toss. We toss again all coins showing tails, and let $X_3$ be the resulting total number of heads , including those left from the previous toss. We continue the process. The pattern is, count heads, toss heads, count tails, toss tails, count heads, toss heads etc... and $X_0=3$. The ${X_n}$ is a Markov Chain. What is the transition probability matrix?
I thought the transition probability matrix would be this.. $$M = \begin{bmatrix} 0&0&0&1\\[0.3em] 0&0&0.5&0.5\\ 0 & (0.5)^2 & 2(0.5)^2 & (0.5)^2 \\[0.3em] (0.5)^3 &3(0.5)^3 &3(0.5)^3 &(0.5)^3\\ \end{bmatrix}$$
where the state space is 0,1,2,3 .. is this correct?