I came up with this task myself so it might be blurry, actually I changed a bit another exercise which was easy, but I'd like to know the way of coming up to an answer if it was like that. Let $X_n$ be a Markov Chain with values in $S=[1,2,3,4]$, let P be a transition matrix of this Markov chain.
$$\\p= \begin{pmatrix} \frac{1}{4} & 0 & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{2}& 0 \\ 0 & \frac{2}{3} & \frac{1}{3}& 0\\ 0 & \frac{1}{2}& 0 & \frac{1}{2} \end{pmatrix}$$
Calculate $EX_{\tau}$ where $\tau=[inf\space n: \space X_n=2 \space or\space X_n=3]$ knowing we start from $X_0=1$
I know how to calculate $E\tau$ but this is quite problematic for me, any hint appreciated.
Hint
$$\mathbb E[X_\tau]=\sum_{n\geq 1}\mathbb E[X_\tau\mid \tau=n]\mathbb P\{\tau=n\}=\sum_{n\geq 1}\sum_{k\in\{2,3\}}k\mathbb P\{X_\tau=k,\tau=n\}.$$
Recall that for $k\in \{2,3\}$, $$\mathbb P\{X_\tau=k, \tau=n\}=\mathbb P\{X_0=1,X_1\in\{1,4\},...,X_{n-1}\in \{1,4\},X_n=k\}$$