Originally, martingale referred to a class of betting strategies popular in 18th century France. The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since a gambler with infinite wealth is guaranteed to eventually flip heads, the martingale betting strategy was seen as a sure thing by those who practiced it. Unfortunately, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would quickly bankrupt those foolish enough to use the martingale after even a moderately long run of bad luck.
Suppose that someone applies the martingale betting system at an American roulette table, with 0 and 00 values; a bet on either red or black will win 18 times out of each 38. If the player’s initial bankroll is $160 and the betting unit is $10, the player will make a win in approximately 96% of sessions, gaining an average of $4.30 from each winning session. In the remaining 4% of sessions, the player will “bust”, exhausting his bankroll, for a loss of $160. It follows then that the average session losses of a gambler employing this strategy are $2.27. Given a larger bankroll, the odds of making a win before running out of cash increase; however, the average winnings grow more slowly than the average losses, so the game remains a losing proposition.
Modern casinos generally have table minimums and maximums to prevent players from doubling their bets more than five or six times, rendering the martingale system obsolete.