Applications of Random Walk and Gambler’s Ruin on Irreducible Periodic Markov Chain

Abstract

The Physical or Mathematical behaviour of any system may be represented by describing all the different states it may occupy and by indicating how it moves among these states. In this study, the states of the Markov chain with the integers 0,±1,±2,. . (the drunkard’s straight line) where the only transitions from any state ???? are to neighbouring states (???? + 1): a step to the right with probability ???? and (???? − 1): a step to the left with probability ???? = (1 − ????) has been investigated, in order to provide some insight in determining whether the gambler is ruined, that is, loses all his money in which Markov chain moves to state 0, and taken to be an absorbing state or wins a fortune that Markov chain moves into absorbing state ???? > ????, where ???? is large). Our quest is to analyse the transition diagram and probability transition matrix to obtain the solution to the system of linear equations for the gambler’s ruin problem. The theorems, Gaussian elimination method with the help of some existing equations and laws in Markov chain are used. Illustrative example is considered on playing cards and the following probabilities are obtained: The probability that Grace ends up with all the cards, the probability that Gloria ends up with all the cards and the probability that Gloria takes all of Grace’s cards.