Let $S_n=\sum_{i=1}^n X_i$ be a random walk on the integers, with negative drift and only upward jumps of height 1.
Hence $ P(X_i \geq 2)=0$ and $ E[X_i]<0$.
Compute the probability that the random walk reaches the positive integer $b$ in finite time.
I does NOT say simple random walk. So $X_i$ can take any negative value with any probability.
By the SLLN and the negative drift we can already deduce, that $S_n \to -\infty$ P-a.s.. Therefore $P(\sup_{n\in N}S_n =+\infty)=0$ and it follows, that $P(\tau_b <\infty)\neq 1$ for some $b$.
But in my opinion it is not possible to compute the probability of $S_n$ reaching $b$ in finite time explicitly without more information on the distribution of $X_i$.
Am i wrong ? Can you compute that probability only with the infos given on $X_i$ ?