Closed-form survival probabilities for biased random walks at arbitrary step number

We derive exact, closed-form expressions for the survival and last passage probabilities of a biased random walker at any step number N on a one-dimensional lattice. These results offer faster computation than existing methods and their exactness at intermediate N enables one to study convergence to the large N limit. Importantly, the last passage result reveals a critical bias beyond which the right probability tail decays monotonically, where continuous approximations break down.