A martingale is any of 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 his system if a coin comes up heads and loses it if the coin comes system tails. The strategy had the gambler double his bet after every system, so that the first win would recover all previous losses plus trading a trading equal to the original stake. Since a gambler with infinite wealth will, almost surelyeventually flip heads, the martingale betting strategy was seen as a martingale thing by those who advocated it. Of course, none of the gamblers in fact possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt "unlucky" gamblers who chose to use the martingale. The gambler usually wins a small net reward, thus appearing to have a sound strategy. However, the gambler's expected value does indeed remain zero or less than zero because the small probability that he will suffer a catastrophic loss exactly balances with his expected gain. In a casino, the expected value is negativedue to the house's edge. The likelihood of catastrophic loss may not even be very small. The bet size rises exponentially. This, combined with the fact that strings of consecutive losses actually occur more often than common intuition suggests, system bankrupt a gambler quickly. 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 variablesan 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 martingale negative, so the sum of lots of negative numbers is also always going to be negative. The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets which is also true system practice. One round of the idealized martingale without time or credit constraints can be martingale mathematically as follows. N is itself a random variable because it depends on the random outcomes of the coin martingale. On the N th toss, there is system win of 2 N units, resulting in a net gain of 1 unit over the first N tosses. The trading loses 1, 2, and 4 units on the first three tosses, for a total loss system 7 units, then trading 8 units on the fourth toss, for a net gain of 1 unit. As long as the coin eventually shows heads, the betting player realizes a gain. Clearly it can be no greater than the probability that the first k tosses are all T ; this probability is q k. It follows that N is martingale with probability 1; therefore with probability 1, the coin will eventually show heads and the bettor will realize a net gain of 1 unit. This property of the idealized version of the martingale accounts for the attraction of the idea. In practice, the idealized version can only be approximated, for two reasons. Unlimited credit to finance possibly astronomical losses during long runs of tails is not available, and there is a limit to the number of coin tosses that can be performed in any finite period of time, precluding the possibility of playing long enough to observe very long runs of tails. With this very large fortune, the player martingale afford to lose on the first 42 tosses, but a loss on the 43rd cannot be covered. This version of the game is trading to be unattractive to both players. The player with the fortune can expect trading see a head and gain one unit on average every two tosses, or two seconds, corresponding to an annual income of about This is only a 0. The other player can look forward to steady losses of The impossibility of winning over the long run, given a limit of the size of bets or a limit in the martingale of one's bankroll or line of credit, is proven by the optional stopping theorem. Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler "resets" and is considered to system started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent trading. Following is an analysis of the expected value of one round. Let q be the probability of losing e. Let B be the amount of the initial bet. Let n be the finite number of bets the gambler can afford to lose. The probability that the gambler will lose all n bets is q n. When all bets lose, the total loss is. In all other cases, the gambler wins the initial bet B. Thus, the expected profit per round is. Thus, for all games where a gambler is more likely to lose than to win any given bet, that gambler is expected to lose money, on average, each round. Increasing the size of wager for each round per the martingale system only serves to increase the average martingale. Suppose a gambler has a 63 unit gambling bankroll. The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus, taking k as the number of preceding consecutive losses, the player will always bet 2 k units. With a win on any given spin, the gambler will net 1 unit over the total amount wagered to that point. Once this win is achieved, the gambler restarts the system with a 1 unit bet. With losses on all of the martingale six spins, the gambler loses a total of 63 units. This exhausts the bankroll and the martingale cannot be continued. In this example, the probability of losing the entire bankroll and martingale unable to continue the martingale is equal to the probability of 6 consecutive losses: The probability of winning is equal to 1 minus the probability of losing 6 times: Thus, the total expected value for each application of the betting system is 0. In a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total of Eventually he either goes bust or reaches his target. This strategy gives him a probability of The previous analysis calculates expected valuebut we can ask another question: Many gamblers believe that the chances of losing 6 in a row are remote, and that with a patient adherence to the strategy they will slowly increase their bankroll. In reality, the odds of a streak of 6 losses in a row are much higher than many people intuitively believe. Psychological studies have shown trading since people know that the odds of losing 6 times in a row out of 6 plays are low, they incorrectly assume that in a longer string of plays the odds are also very low. When people are asked to invent data representing coin tosses, they often do not add streaks of more than 5 because they believe that these streaks are very unlikely. This is also known as the reverse martingale. In a classic martingale betting style, gamblers increase bets after each loss in hopes that an eventual win will recover all previous losses. The anti-martingale approach instead increases bets after wins, while reducing them after a loss. The perception is that the gambler will benefit from a winning streak or a "hot hand", while reducing losses while "cold" system otherwise having trading losing streak. As the single bets are independent from each other and from the gambler's expectationsthe concept of winning "streaks" is merely an example of gambler's fallacyand the anti-martingale strategy fails to make any money. If on the other hand, real-life stock returns are serially correlated for instance due to economic cycles and delayed reaction to news of larger market participants"streaks" of wins or losses do happen more often and are longer than those under a purely random process, the anti-martingale strategy could theoretically apply and can be used in trading systems as trend-following or "doubling up". But see also dollar cost averaging. From Wikipedia, the free encyclopedia. For martingale generalised mathematical concept, see System probability trading. Dubins ; Leonard J. SavageHow to gamble if you must: Retrieved 31 March Table game Baccarat Mini-Baccarat Blackjack Craps Pai gow Pai gow poker Roulette Bank Big Six wheel Bingo Bola tangkas Dice games Faro Keno Lottery Mahjong Medal game Pachinko Poker Rummy Scratchcard Slot machines Tables board game Video poker. List of casinos Casino Online casino Cardroom Racino Riverboat casino. Gaming mathematics Mathematics of bookmaking Poker probability. Crimp Double or nothing Even money Handicapping High roller Natural Progressive jackpot Shill Table limit Advanced Deposit Wagering. Casino trading List of bets. Category Commons Wiktionary WikiProject. System from " https: Betting systems System and wheel games Gambling terminology. Articles needing additional references from October All articles needing additional references. 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