On a Variation of Gambler's Ruin Problem

TL;DR

This paper analyzes a variation of the gambler's ruin problem where the gambler's gains and losses depend on the occurrences of specific words in a Markov chain text, providing probabilities and expected waiting times for certain outcomes.

Contribution

It introduces a novel approach to gambler's ruin by incorporating Markov chain-based word occurrences and derives formulas for probabilities and expected times.

Findings

Derived probability formulas for reaching gain or loss thresholds.

Calculated expected waiting times until an outcome occurs.

Extended gambler's ruin analysis to Markov chain text models.

Abstract

Assume that letters (from a finite alphabet) in a text form a Markov chain. We track two distinct words, $U$ and $D$. A gambler gains 1 point for each occurrence of $U$ (including overlapping occurrences) and loses 1 point for each occurrence of $D$ (also including overlapping occurrences). We determine the probability of gaining $A$ points before losing $B$ points, where $A$ and $B$ are integers. Additionally, we find the expected waiting time until one of the two events -- gaining $A$ points or losing $B$ points -- occurs.