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

We present a closed-form expression for the survival probability of a biased random walker to first reach a target site on a one-dimensional lattice. The expression holds for any step number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math> and is computationally faster than non-closed-form results in the literature. Because our result is exact even in the intermediate step number range, it serves as a tool to study convergence to the large <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>N</b:mi></b:math> limit. We also obtain a closed-form expression for the probability of last passage. In contrast to predictions of the large <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>N</c:mi></c:math> approximation, the new expression reveals a critical value of the bias beyond which the tail of the last-passage probability decays monotonically.