  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.
