Words in Random Binary Sequences I

TL;DR

This paper derives recurrence relations and generating functions to analyze the probabilities of specific binary words appearing for the first time in sequences generated by fair coin flips.

Contribution

It introduces second- and third-order recurrence relations and generating functions for first occurrence probabilities of binary words of length 2 or 3.

Findings

Derived explicit recurrence relations for first occurrence probabilities.

Established generating functions for these probabilities.

Provided analytical tools for studying pattern appearances in random sequences.

Abstract

When flipping a fair coin, let $W = L_1L_2...L_N$ with $L_i\in\{H,T\}$ be a binary word of length $N=2$ or $N=3$. In this paper, we establish second- and third-order linear recurrence relations and their generating functions to discuss the probabilities $p_{W}(n)$ that binary words $W$ appear for the first time after $n$ coin tosses.