Gambler's Ruin? Some Aspects of Coin Tossing

TL;DR

This paper analyzes the expected number of coin tosses needed for specific sequences to appear, introducing a new algorithm that simplifies calculations for longer sequences and reveals that these averages are always even integers.

Contribution

It presents a novel algorithm for calculating the expected number of tosses for any sequence, simplifying the process even for complex strings.

Findings

Average number of tosses is always an even integer.

Specific sequences like tail-head and two heads have different expected lengths.

The new algorithm enables simple hand calculations for complex sequences.

Abstract

What is the average number of tosses needed before a particular sequence of heads and tails turns up? We solve the problem didactically, starting with doubles, finding that a tail, followed by a head, turns up on the average after only four tosses, while six tosses are needed for two successive heads. The method is extended to encompass the triples head-tail-tail and head-head-tail, but head-tail-head and head-head-head are surprisingly more recalcitrant. However, the general case is finally solved by a new algorithm that allows a simple computation that can be done by hand, even for relatively long strings. It is shown that the average number of tosses is always an even integer.