In the drunkard's walk problem, a person performs a random walk on the integers ${0,1,2,\ldots , n, n+1}$, beginning at the integer $1 \leq i \leq n$. At every time-step the drunkard takes one step right with probability $p$ or one step left with probability $q$ (s.t. $p+q=1)$. When the drunkard reaches either $0$ or $n+1$ we say he has "fallen off the cliff", and the process terminates.
What is the probability that it takes more than $m$ steps for the drunk man to fall off? Asymptotics are also welcome. Please also give a reference text for this problem if you know of any!