TITLE:
Competing Patterns in Bernoulli Sequence of Trials
AUTHORS:
Paul Vrbik, Jan Vrbik
KEYWORDS:
Competing Patterns, Bernoulli Sequence of Trials, Game Theory
JOURNAL NAME:
Applied Mathematics,
Vol.14 No.5,
May
29,
2023
ABSTRACT: Consider performing a sequence of Bernoulli trials (each resulting in either a success, denoted S, or a failure F, with a probability of p and q := 1 - p respectively) until one of m specific strings (or patterns) of consecutive outcomes is generated. This can be seen as a game where m players select one such pattern each and the one whose pattern occurs first wins. We present symbolic formulas for the m probabilities of winning, and for the mean number of trials and the corresponding standard deviation to complete this game. Several numerical examples are presented, including a search for optimal strategy.