Suppose I have a 1D random walk with random variables $X_i \in \left\{1, -1 \right\}$, and $p = \Pr{\left(X_i = 1\right)}$ and $q = 1-p = \Pr{\left(X_i = -1\right)}$. Let $S_n = S_0 + \sum_{i=1}^n X_i$ be the current position in the walk. From example 1.6.1 on this page, I know that if $p \neq q$, the walk is transient, which means that if the walk begins at the origin, it will not revisit the origin infinitely many times. We can understand this by noting that one of the probabilities of moving to the left or right is larger, so eventually the walk will "drift" away.
But I'm interested in knowing what is the time $T$ such that we revisit the origin $c$ times with probability $r$. In general I'd like to know what happens when we start at some arbitrary point $x_0$ and revisit a point $x$, but I'm happy with looking at the case $x_0 = x = 0$.
In other words, how long should I expect to wait (the time $T$), such that I can be confident (probability $r$) that the walk will revisit the origin $c$ times?
I know a little bit about random walks, but not a lot, so any pointers or strategies are appreciated.
Update: I started by finding the probability $P$ of the random walk returning to the origin once after $T$ steps, and then took the estimated probability of having $c$ returns as $P^c$ (since the steps in the random walk are independent events), but this is a drastic underestimate of the actual probability (based on some numerical experiments I conducted).