532:: at each spin, he should bet the smallest amount such that if he wins he reaches his target immediately, and if he does not have enough for this, he should simply bet everything. Eventually he either goes bust or reaches his target. This strategy gives him a probability of 97.8744% of achieving the goal of winning one unit vs. a 2.1256% chance of losing all 63 units, and that is the best probability possible in this circumstance. However, bold play is not always the optimal strategy for having the biggest possible chance to increase an initial capital to some desired higher amount. If the gambler can bet arbitrarily small amounts at arbitrarily long odds (but still with the same expected loss of 10/19 of the stake at each bet), and can only place one bet at each spin, then there are strategies with above 98% chance of attaining his goal, and these use very timid play unless the gambler is close to losing all his capital, in which case he does switch to extremely bold play.
594:
184:, 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.
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140:. 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
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means the bettor must have over 65,500 (2^15-1 for their 15 losses and 2^15 for their 16th streak-ending winning bet) times their original bet size. Thus, a player making 10 unit bets would want to have over 655,000 units in their bankroll (and still have a ~5.5% chance of losing it all during 5,000 plays).
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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, also known as the reverse martingale, instead increases bets after wins, while reducing them after a loss. The perception is that
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during a string of 200 plays is approximately 84%. Even if the gambler can tolerate betting ~1,000 times their original bet, a streak of 10 losses in a row has an ~11% chance of occurring in a string of 200 plays. Such a loss streak would likely wipe out the bettor, as 10 consecutive losses using the
219:
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 have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds.
159:
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
551:
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 that 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
179:
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
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These unintuitively risky probabilities raise the bankroll requirement for "safe" long-term martingale betting to infeasibly high numbers. To have an under 10% chance of failing to survive a long loss streak during 5,000 plays, the bettor must have enough to double their bets for 15 losses. This
547:
As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we are betting red/black or even/odd. 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.
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the gambler will benefit from a winning streak or a "hot hand", while reducing losses while "cold" or otherwise having a losing streak. As the single bets are independent from each other (and from the gambler's expectations), the concept of winning "streaks" is merely an example of
497: > 0. 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 loss.
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In this example, the probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 consecutive losses: (10/19) = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times:
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However, without these limits, the martingale betting strategy is certain to make money for the gambler because the chance of at least one coin flip coming up heads approaches one as the number of coin flips approaches infinity.
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When people are asked to invent data representing 200 coin tosses, they often do not add streaks of more than 5 because they believe that these streaks are very unlikely. This intuitive belief is sometimes referred to as the
155:, 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
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also very low. In fact, while the chance of losing 6 times in a row in 6 plays is a relatively low 1.8% on a single-zero wheel, the probability of losing 6 times in a row (i.e. encountering a streak of 6 losses)
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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 in practice). It is only with unbounded wealth, bets
181:
544:, but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll?
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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.
164:. Additionally, as the likelihood of a string of consecutive losses is higher than common intuition suggests, martingale strategies can bankrupt a gambler quickly.
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The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the
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With losses on all of the first six spins, the gambler loses a total of 63 units. This exhausts the bankroll and the martingale cannot be continued.
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In a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total of 64. Assuming
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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
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will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected value is negative, due to the
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Thus, the total expected value for each application of the betting system is (0.978744 − 1.339118) = −0.360374 .
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Bold play and the optimal policy for Vardi's casino, pp 150–156 in: Random Walk, Sequential
Analysis and Related Topics
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be the probability of losing (e.g. for
American double-zero roulette, it is 20/38 for a bet on black or red). Let
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Gambling strategy where the amount is raised until a person wins or becomes insolvent
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as the number of preceding consecutive losses, the player will always bet 2 units.
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677:"Noise, Information, and the Favorite-Longshot Bias in Parimutuel Predictions"
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Probability and computing: randomized algorithms and probabilistic analysis
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798:"What were the Odds of Having Such a Terrible Streak at the Casino?"
475:{\displaystyle (1-q^{n})\cdot B-q^{n}\cdot B(2^{n}-1)=B(1-(2q)^{n})}
628: – Cost that has already been incurred and cannot be recovered
823:"Martingale Strategy and Averaging Down | What You Need to Know"
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How to gamble if you must: inequalities for stochastic processes
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martingale strategy means a loss of 1,023x the original bet.
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be the finite number of bets the gambler can afford to lose.
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Following is an analysis of the expected value of one round.
622: – Paradox involving a game with repeated coin flipping
582:, and the anti-martingale strategy fails to make any money.
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time that it could be argued that the martingale becomes a
348:. In all other cases, the gambler wins the initial bet (
330:{\displaystyle \sum _{i=1}^{n}B\cdot 2^{i-1}=B(2^{n}-1)}
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independent and identically distributed random variables
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Pages displaying short descriptions of redirect targets
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The expected amount lost is (63 × 0.021256)= 1.339118.
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The expected amount won is (1 × 0.978744) = 0.978744.
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that originated from and were popular in 18th-century
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675:Ottaviani, Marco; Sørensen, Peter Norman (2010).
167:The martingale strategy has also been applied to
238:The probability that the gambler will lose all
340:The probability the gambler does not lose all
19:For the generalised mathematical concept, see
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514:1 − (10/19) = 97.8744%.
723:, Cambridge University Press, p. 298,
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648:Williams, Leighton Vaughan (2021-09-15).
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116:Learn how and when to remove this message
717:Michael Mitzenmacher; Eli Upfal (2005),
246:. When all bets lose, the total loss is
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215:Mathematical analysis of a single round
231:be the amount of the initial bet. Let
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54:adding citations to reliable sources
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796:Martin, Frank A. (February 2009).
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540:The previous analysis calculates
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1336:Video game addiction
946:Gambling mathematics
882:Gambling terminology
778:, World Scientific,
772:Larry Shepp (2006),
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356:profit per round is
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50:improve this article
737:on October 13, 2015
996:Due Column betting
974:Advantage gambling
803:. WizardOfOdds.com
600:Mathematics portal
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157:exponential growth
134:betting strategies
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39:This article
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1005:
986:Dice control
830:. Retrieved
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805:. Retrieved
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735:the original
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1006:Martingale
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832:2024-03-11
636:References
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701:25760376
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169:roulette
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