The following is a passage from the lecture notes:
Let a simple random move to the right with probability $p$ and to the left with probability $q = 1 − p$. We want the probability that it hits level $b$ before it hits level $c$, started at $x$, $b <x< c$. Let $τ$ be the stopping time of random walk hitting $b$ or $c$. Then, Stopping the exponential martingale $M_n = (\frac{q}{p})^{X_n}$ we have $\mathbb{E}(\frac{q}{p})^{X_τ} = (\frac{q}{p})^x \dots$
I do not understand why they take an exponential martingale.