Let $p$ be a probability in $]0,1[$, and let $(X^p_i)_{i \geq 1}$ be a i.i.d. family of variables with law $P(X=1)=p, P(X=-\frac{p}{1-p})=1-p$ (so that $E(X)=0$). Set $S^p_n=\sum_{k=1}^{n} X^p_k$ for $n\geq 1$, let $e(p,n)$ denote the probability that at least one of $S^p_1,S^p_2, \ldots ,S^p_n$ is positive (which means that the random walk reaches a positive value at least once during the first $n$ steps ). It is well known that for a fixed $p$, $e(p,n) \to 1$ when $n\to +\infty$.

Let $p\neq q$ be two probabilities. For large enough $n$, shall we have $e(n,p) \lt e(n,q)$ or $e(n,p) \gt e(n,q)$ ?