Gambler's ruin

The term gambler's ruin is a statistical concept expressed in a variety of forms: The concept has specific relevance for gamblers; however it also leads to mathematical theorems with wide application and many related results in probability and statistics. Huygens' result in particular led to important advances in the mathematical theory of probability. The earliest known mention of the gambler's ruin problem is a letter from Blaise Pascal to Pierre Fermat in 1656 (two years after the more famous correspondence on the problem of points). Pascal's version was summarized in a 1656 letter from Pierre de Carcavi to Huygens: Huygens reformulated the problem and published it in De ratiociniis in ludo aleae ('On Reasoning in Games of Chance', 1657): This is the classic gambler's ruin formulation: two players begin with fixed stakes, transferring points until one or the other is 'ruined' by getting to zero points. However, the term 'gambler's ruin' was not applied until many years later. Let 'bankroll' be the amount of money a gambler has at his disposal at any moment, and let N be any positive integer. Suppose that he raises his stake to bankroll N {displaystyle {frac { ext{bankroll}}{N}}} when he wins, but does not reduce his stake when he loses. This general pattern is not uncommon among real gamblers, and casinos encourage it by 'chipping up' winners (giving them higher denomination chips). Under this betting scheme, it will take at most N losing bets in a row to bankrupt him. If his probability of winning each bet is less than 1 (if it is 1, then he is no gambler), he will eventually lose N bets in a row, however big N is. It is not necessary that he follow the precise rule, just that he increase his bet fast enough as he wins. This is true even if the expected value of each bet is positive.