A note on the gambling team method

Abstract

Gerber and Li in [4] formulated, using a Markov chain embedding, a system of equations that describes relations between generating functions of waiting time distributions for occurrences of patterns in a sequence of independent repeated experiments when initial outcomes of the process are known. We show how this system of equations can be obtained by using the classical gambling team technique . We also present a form of solution of the system and give an example showing how first results of trials influence the probabilities that a chosen pattern precedes remaining ones in a realization of the process.2010 Mathematics Subject Classification: 60G40, 60G42.

Penney’s game between many players

“…It leads to considerations of the game with an initial pattern (compare Li [6, the system of equations (3.7)]). A form of solutions of that system presented in [9,Cor.4.3] may also be treated as some another generalization of the Conway's formula to the case when an initial pattern is known.…”