The "Monkey Typing Shakespeare" Problem for Compositions

TL;DR

This paper explores a probabilistic problem involving candy-eating schedules, analyzing the likelihood of avoiding specific three-day consecutive patterns, and provides algorithms implemented in Maple for such combinatorial questions.

Contribution

It introduces algorithms for calculating probabilities of complex patterns in constrained sequences, implemented in Maple packages, addressing a novel combinatorial problem.

Findings

Algorithms for pattern avoidance probabilities

Implementation in Maple packages

Applicable to generalized sequence problems

Abstract

Suppose that your mother gave you n candies. You have to eat at least one candy each day. One possibility is to eat all n of them the first day. The other extreme is to make them last n days, and only eat one candy a day. Altogether, you have, famously, 2 to the power n-1 choices. If each such choice is equally likely, what is the probability that you never have three consecutive days, where in the first day you ate at least 2 candies, in the second day you ate at least 5 candies, and in the third day you ate at least 3 candies? This article describes algorithms, fully implemented in two Maple packages, to answer such important questions, and more general ones, of this kind.