Let $(X_i)_{i\le n}\sim B(p)$ be a length $n$ sequence of iid. Bernoulli random variables. That is $\Pr[X_i=1]=p$ and $\Pr[X_i=0] = 1-p$. Let $S_k = \sum_{i\le k} X_i$ be the sum of the first $k$ variables. What is the probability that each $S_k \ge p k$? That is the probability that the random walk $S_k-pk$ stays non-negative?
Clearly, the answer must be at most $p$, since the first step has to succeed. That is $X_1=1$. For $p<<1/k$, this would also be a lower bound, since we would only expect one of the $X_i$ to ever be positive.
From comparison with Brownian motion, we might expect the probability tot be around $1/\sqrt{n}$.
I wonder if one could prove that these are indeed the two cases to look at. That is the probability that for all $k\le n$, $S_k \ge pk$ is $\sim p/\sqrt{k}$. It seems like a problem that has been well studied, but I couldn't find anything in e.g. Feller except for $p=1/2$.
A very related question is requiring that $S_k \ge \lfloor pk\rfloor$. It appears that here the success probability is always at least $1/\sqrt k$. That makes sense since the above version requires $X_1=1$ and then continues very similarly to the floored version.
Update: I realized I could plot the probability for various $p$ and $k$.
As @antkam says the plot is discontinuous roughly at $p=1/n$ for all integers $1<n<k$. However it seems to be pretty smooth asymptotically.
It also seems like a lower bound of $C p/\sqrt{k}$ should be achievable for some constant $0<C<1$, while an upper bound would have to go through (1,1).
Here thee blue line is the true probability (as calculated by dynamic programming) while the orange line is $p/\sqrt{k}$.