On the duration of a gambler's ruin problem

Abstract

Consider a gambler who on each bet either wins 1 with probability p or loses 1 with probability q=1−p, with the results of successive bets being independent. The gambler will stop betting when they are either up k or down k. Letting N be the number of bets made, we show that N is a new better than used random variable. Moreover, we show that if k is even then N/2 has an increasing failure rate, and if k is odd then (N+1)/2 has an increasing failure rate.