I'm working on a problem from Adventures in Stochastic Processes by Sidney Resnick (2.19c). In short, we are given the following transition matrix: $$P=\begin{bmatrix} 0.6 & 0.1 & 0.1 & 0.1 & 0.1 & 0 \\ 0.3 & 0 & 0.3 & 0.2 & 0.1 & 0.1 \\ 0.3 & 0.5 & 0 & 0.1 & 0.1 & 0 \\ 0.1 & 0.2 & 0.2 & 0 & 0.1 & 0.4 \\ 0.2 & 0.2 & 0 & 0.1 & 0 & 0.5 \\ 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0\end{bmatrix}$$ We assume the Markov chain begins at state $3$, and the state space is $\{1, 2, \ldots, 6\}$. We are asked to determine the expected number of transitions until state $6$ is visited 5 times.
Let $E_I(x)$ be the expected number of transitions for the Markov Chain to reach $x$ if it starts at $I$. My thought was that this can be computed as $E_3(6) + 5E_6(6)$, as the expected number of transitions to visit $6$ from $3$, and then the expected number of transitions to revisit $6$. Is this correct?
EDIT (2021 September 9): As pointed out by @Graham Kemp in the comments, my initial formulation is incorrect. We are interested in the expected number of transitions to $3$ to $6$, and the number of transitions from $6$ back to $6$. As we require $5$ total transitions, we have 1 from $3$ to $6$ and then $ 5 - 1 = 4$ transitions from $6$ to $6$. This should thus instead be $E_3(6) + 4E_6(6)$.