A martingale is a class of betting strategies that originated from and were popular in 18th-century France. The simplest of 🛡 these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads 🛡 and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that 🛡 the first win would recover all previous losses plus win a profit equal to the original stake. Thus the strategy 🛡 is an instantiation of the St. Petersburg paradox.

Since a gambler will almost surely eventually flip heads, the martingale betting strategy 🛡 is certain to make money for the gambler provided they have infinite wealth and there is no limit on money 🛡 earned in a single bet. However, no gambler has infinite wealth, and the exponential growth of the bets can bankrupt 🛡 unlucky gamblers who choose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins 🛡 a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains zero because the small 🛡 probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected 🛡 value is negative, due to the house's edge. Additionally, as the likelihood of a string of consecutive losses is higher 🛡 than common intuition suggests, martingale strategies can bankrupt a gambler quickly.

The martingale strategy has also been applied to roulette, as 🛡 the probability of hitting either red or black is close to 50%.

Intuitive analysis [ edit ]

The fundamental reason why all 🛡 martingale-type betting systems fail is that no amount of information about the results of past bets can be used to 🛡 predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption 🛡 that the win–loss outcomes of each bet are independent and identically distributed random variables, an assumption which is valid in 🛡 many realistic situations. It follows from this assumption that the expected value of a series of bets is equal to 🛡 the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet 🛡 times the probability that the player will make that bet. In most casino games, the expected value of any individual 🛡 bet is negative, so the sum of many negative numbers will also always be negative.