The conditional gambler’s ruin problem with ties allowed

Abstract

We determine the distribution of duration in the gambler’s ruin problem given that one specific player wins. In this version we allow ties in the single games. We present a unified approach which uses generating functions to prove and extend some results that were obtained in [Frederick Stern, Conditional expectation of the duration in the classical ruin problem, Math. Mag. 48 (4) (1975) 200–203; S.M. Samuels, The classical ruin problem with equal initial fortunes, Math. Mag. 48 (5) (1975) 286–288; W.A. Beyer, M.S. Waterman, Symmetries for conditioned ruin problems, Math. Mag. 50 (1) (1977) 42–45].