On the Probability of Gambler’s Ruin System

Abstract:

A typical Gambler’s ruin problem was considered in this work by applying theory of difference equations. The expected gains and duration of the game were generated for both fair and unfair games. In order to develop the probabilities that a Gambler meets the target of initial capital units, the frequency on steps hitting zero or initial capital was computed. As a particular unit tends to infinity, the sum of probabilities of ruin and success was established to be unity. Boundary conditions were set up to develop the generating function for the duration of the game. To ascertain these, the conditions were validated by finding the generating functions for probability of gambler’s ruin at zero and unit absorptions. The illustrated example showed that the expected gain is 0 when the game is fair and negative when it is unfair. While the expected duration of the game was found to be infinite for a fair game, an infinitely prolonged game series is infeasible.