Asian Journal of Probability and Statistics

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.