In the early years of the eighteenth century A. De Moivre, N. Bernoulli, and P. R. Montmort found three solutions for the following problem of games of chance: Two players, A and B, play a series of games. In each game, independently of the others, either A wins a counter from B with probabilitypor B wins a counter from A with probabilityq(p+q= 1). The series ends if either A wins a total number ofacounters from B or B wins a total number ofbcounters from A. What is the probability that A wins the series in at mostngames? Denote this probability byPn(a,b). In this paper simple and elementary proofs are given for the various formulas forPn(a,b). Furthermore, it is shown how these formulas can be applied in the theories of order statistics, random walks, storage, queues, Brownian motion, and dams.