Gambler's Ruin? Some Aspects of Coin Tossing

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.