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A martingale is a class of betting strategies that originated from and were popular in 18th-century France. The simplest of 4️⃣ these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads 4️⃣ and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that 4️⃣ the first win would recover all previous losses plus win a profit equal to the original stake. Thus the strategy 4️⃣ is an instantiation of the St. Petersburg paradox.

Since a gambler will almost surely eventually flip heads, the martingale betting strategy 4️⃣ is certain to make money for the gambler provided they have infinite wealth and there is no limit on money 4️⃣ earned in a single bet. However, no gambler has infinite wealth, and the exponential growth of the bets can bankrupt 4️⃣ unlucky gamblers who choose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins 4️⃣ a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains zero because the small 4️⃣ probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected 4️⃣ value is negative, due to the house's edge. Additionally, as the likelihood of a string of consecutive losses is higher 4️⃣ than common intuition suggests, martingale strategies can bankrupt a gambler quickly.

The martingale strategy has also been applied to roulette, as 4️⃣ the probability of hitting either red or black is close to 50%.

Intuitive analysis [ edit ]

The fundamental reason why all 4️⃣ martingale-type betting systems fail is that no amount of information about the results of past bets can be used to 4️⃣ predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption 4️⃣ that the win–loss outcomes of each bet are independent and identically distributed random variables, an assumption which is valid in 4️⃣ many realistic situations. It follows from this assumption that the expected value of a series of bets is equal to 4️⃣ the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet 4️⃣ times the probability that the player will make that bet. In most casino games, the expected value of any individual 4️⃣ bet is negative, so the sum of many negative numbers will also always be negative.