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. 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. Le Chiffre then goes to the casino Royale-les-Eaux in an attempt to replace his lost funds. MI6 sends Bond, an expert baccarat player, to the casino to bankrupt Le Chiffre and force him to take refuge with the British government and inform on SMERSH. Bond bests Le Chiffre in a game of Chemin de Fer, taking all of his money. Technically, whether it is on the floor, credits in the machine, or a ticket hanging out, it belongs to the casino. I believe the law may even say that. And if you even look like you are searching for money or machine credits, you can even be thrown out for that. That said, I have found money, and kept it. How to recover your Google Account or Gmail. Tips to complete account recovery steps. Secure a hacked or compromised Google Account. Recover a recently deleted Google Account. Why your account recovery request is delayed. Change or reset your password. Can't sign in to your Google Account.

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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 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.

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 possess infinite wealth, and the exponential growth of the bets can bankrupt unlucky gamblers who chose 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 occurs more often than common intuition suggests, martingale strategies can bankrupt a gambler quickly.

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

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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.

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).^[1] It is only with unbounded wealth, bets and time that it could be argued that the martingale becomes a winning strategy.

Mathematical analysis[edit]

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 optional stopping theorem.^[1]

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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.

Mathematical analysis of a single round[edit]

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. Following is an analysis of the expected value of one round.

Let q be the probability of losing (e.g. for American double-zero roulette, it is 20/38 for a bet on black or red). 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 qⁿ. When all bets lose, the total loss is

{displaystyle sum _{i=1}^{n}Bcdot 2^{i-1}=B(2^{n}-1)}

The probability the gambler does not lose all n bets is 1 − qⁿ. In all other cases, the gambler wins the initial bet (B.) Thus, the expected profit per round is

{displaystyle (1-q^{n})cdot B-q^{n}cdot B(2^{n}-1)=B(1-(2q)^{n})}

Whenever q > 1/2, the expression 1 − (2q)ⁿ < 0 for all n > 0. Thus, for all games where a gambler is more likely to lose than to win any given bet, that g

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