A random walker moves to $+1$ with probability $p$ and moves to $-1$ with probability $q=1-p$. If he starts at point $m$, what is the probability that he doesn't hit the point zero after $k$ steps, asymptotically for large $k$ and $m$?
I have found results for the case $p=q=1/2$ for brownian motion in Wiki. The survival probability, the probability that the particle has remained at a position $x < x_c$ for all times up to $t$, is given by
$$ S(t)=\operatorname{erf}\left(\frac{x_c}{2\sqrt{D t}}\right)$$
How can I convert it to the case that $p\neq 1/2$?