A coin flip game and generalizations of Fibonacci numbers
Jia Huang
TL;DR
This work analyzes the waiting time until the first occurrence of a fixed string $S$ in fair coin flips. By conditioning on initial runs and developing recursive relations for $E(S)$, it obtains exact formulas for terminating strings with up to four maximal runs or alternating patterns, including $E(H^k)=\!2^{k+1}-2$, $E(H^kT^\ell)=\!2^{k+\ell}$, and explicit three- and four-run expressions such as $E(H^kT^\ell H^m)=\!2^{k+\ell+m}+2^{\min\{k,m\}+1}-2$ and a two-case formula for $E(H^kT^\ell H^mT^d)$. The authors then link these stopping-time results to generalized Fibonacci numbers, introducing $F_n^k$, $\bar F_n^k$, and two-parameter families $F_n^{k,m}$ and $\tilde F_n^{k,m}$ to express $E_n(S)$ for more complex endings, with several summation identities derived via generating functions. They further analyze alternating ending strings and propose conjectures for arbitrary strings, highlighting symmetry properties and possible extensions to dice and broader probabilistic settings. The results reveal deep connections between stopping times in coin-flip sequences and generalized Fibonacci structures, with potential implications for combinatorial enumeration and probabilistic identities.
Abstract
We study a game in which one keeps flipping a coin until a given finite string of heads and tails occurs. We find the expected number of coin flips to end the game when the ending string consists of at most four maximal runs of heads or tails or alternates between heads and tails. This leads to some summation identities involving certain generalizations of the Fibonacci numbers.