ABSTRACT
Consider a game of chance between two players: A, the gambler and B, the opponent. It is assumed that at each play, A either wins one unit from B with probability p or loses one unit to B with probability q = 1 - p $ q=1-p $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math2_1.tif"/> . Conversely, B either wins from A or loses to A with probabilities q or p. The result of every play of the game is independent of the results of previous plays. The gambler A and the opponent B each start with a given number of units and the game ends when either player has lost his or her initial stake. What is the probability that the gambler loses all his or her money or wins all the opponent’s money, assuming that an unlimited number of plays are possible? This is the classic gambler’s ruin problem 1 . In a simple example of gambler’s ruin, each play could depend on the spin of a fair coin, in which case p = q = 1 2 $ p=q=\frac{1}{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math2_2.tif"/> . The word ruin is used because if the gambler plays a fair game against a bank or casino with unlimited funds, then the gambler is certain to lose.
