The conditional gambler’s ruin problem with ties allowed

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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].

Keywords

Gambler’s ruin
Generating function
Lattice path combinatorics
Continued fraction representation by polynomials
Chebyshev polynomials of the second kind

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