Let $Y_1,..,Y_n$ random variables i.i.d. with $\mathbb{P}[Y_1=1]=p$ and $\mathbb{P}[Y_1=-1]=q$ and $Y_0=0$, consider the random walk $X_n=\sum_{i=0}^nY_i$ and the stopping time \begin{equation} \tau=\inf\{n\in\mathbb{N}| X_n=-a\,\,or\,\, X_n=b\} \end{equation} where $a,b$ are some positive integers. I am trying to compute $\mathbb{P}[X_\tau=b]$ using the fact that $X_n-(p-q)n$ is a martingale respect to the natural filtration of $X_n$. My question is using the optimal sttoping time theorem I got \begin{equation*} 0=\mathbb{E}[X_0-0]=\mathbb{E}(X_\tau-(p-q)\tau)\implies\mathbb{E}(X_\tau)=(p-q)\mathbb{E}(\tau) \end{equation*} and the other hand \begin{equation*} \mathbb{E}(X_\tau)=b\mathbb{P}[X_\tau=b]-a\mathbb{P}[X_\tau=-a]=(b+a)\mathbb{P}[X_\tau=b]-a \end{equation*} Hence \begin{equation*} \mathbb{P}[X_\tau=b]=\frac{(p-q)\mathbb{E}(\tau)+a}{b+a} \end{equation*} but I don't know if I am wrong, or how can compute in this case $\mathbb{E}(\tau)$. I know that is possible to compute $\mathbb{E}(\tau)$ using a martingale $(q/p)^{X_n}$ and Wall's lemma. But is it possible to compute it without these tools?
Any help, advice or hint is welcome. Thank you!.