https://journalajpas.com/index.php/AJPAS <p style="text-align: justify;"><strong>Asian Journal of Probability and Statistics</strong> <strong>(ISSN: 2582-0230) </strong>aims to publish high-quality papers (<a href="https://journalajpas.com/index.php/AJPAS/general-guideline-for-authors">Click here for Types of paper</a>) in all areas of ‘Probability and Statistics’. By not excluding papers based on novelty, this journal facilitates the research and wishes to publish papers as long as they are technically correct and scientifically motivated. The journal also encourages the submission of useful reports of negative results. This is a quality controlled, OPEN peer-reviewed, open-access INTERNATIONAL journal.</p> Asian Journal of Probability and Statistics en-US Asian Journal of Probability and Statistics 2582-0230 https://journalajpas.com/index.php/AJPAS/article/view/939 <p>This study examines alternative methods for solving selected first-passage-time problems in one-dimensional random walks. A simple random walk is considered in which movement to the right occurs with probability p and movement to the left with probability q=1p. The study focuses on first-passage-time probabilities, expected values, variances, and the jth passage time. Its central approach is a proposed counting formula linked to Catalan numbers and related combinatorial identities. The counting structure is used to represent the probability of reaching x=1 for the first time after an odd number of steps and to obtain a Catalan-number form for the first passage-time distribution. The manuscript further derives the expected value and variance of the first passage time using two approaches,! including! generating-function arguments and the proposed counting method. In addition, counted coefficients for successive passage times are examined and used to formulate a general expression for the jth passage time through a Catalan-Ballot convolution identity. The results illustrate a combinatorial connection between first-passage-time probabilities and Catalan-number structures while providing an alternative route to quantities that have also been approached by difference equations, conditioning, generating functions, and related probabilistic methods. The analysis is restricted to one-dimensional random walks, and the direct applicability of the proposed counting formula to higherdimensional walks is not established.</p> Felgona Tana Omondi Fredrick Onyango Isaac Owino Copyright (c) 2026 Author(s). The licensee is the journal publisher. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2026-08-19 2026-08-19 28 9 1 11 10.9734/ajpas/2026/v28i9939