Solving Flipping Coin Problem and Penney-Ante Game without Using Generating Function

We revisit the Flipping Coins (FCP) Problem and PennyAnte Game(PAG) and present a new method for solving them. In FCP and PAG, we consider the process where a string S is generated character by character randomly until certain fixed pattern occurs. FCP asks the expected length of the S while PAG asks the probability of occurrence for each pattern. The classical way (see [13]) of computing those statistic quantities was based on generating function(g-f) with elegant formulas for solving FCP and PAG. Suppose there are t patterns and their total length is L. It requires O(t+t·L) running time, in which O(t) is for solving t variables linear equations, and O(t · L) is for computing the coefficients. We give a more intuitive way to solve FCP and PAG. Our algorithm is based on the Markov-chain model, without using any g-f techniques. It combines several basic algorithms such as KMP and DFS, and its running time is also O(t + t · L). The accuracy of the answer is better then the previous method.