Consider the iid random variables $(X_i)_i$ on $\{-1,1\}$, such that $P[X_i=-1]=p>\frac12$. Then we can define the biased random walk starting at $0$ as
$S_0 := 0$ and $S_i := X_1 + \dots + X_i$. We define the hitting time $$\tau := \inf\{n\ |\ S_n = -1\}.$$
I am interested in the quantity $P[\tau \geq n]$. If possible I would like to find an exponential bound on this quantity, (or necessary conditions for such a bound). Thank you!