A martingale is a class of betting strategies that originated from and were popular in 18th-century France. The simplest of π these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads π and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that π the first win would recover all previous losses plus win a profit equal to the original stake. Thus the strategy π is an instantiation of the St. Petersburg paradox.
Mathematical analysis [ edit ]
Let one round be defined as a sequence of π consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler "resets" and is π considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence π of independent rounds. Following is an analysis of the expected value of one round.
( 1 β q n ) β
π B β q n β
B ( 2 n β 1 ) = B ( 1 β ( 2 q π ) n ) {\displaystyle (1-q^{n})\cdot B-q^{n}\cdot B(2^{n}-1)=B(1-(2q)^{n})}
These unintuitively risky probabilities raise the bankroll requirement for "safe" long-term martingale betting to π infeasibly high numbers. To have an under 10% chance of failing to survive a long loss streak during 5,000 plays, π the bettor must have enough to double their bets for 15 losses. This means the bettor must have over 65,500 π (2^15-1 for their 15 losses and 2^15 for their 16th streak-ending winning bet) times their original bet size. Thus, a π player making 10 unit bets would want to have over 655,000 units in their bankroll (and still have a ~5.5% π chance of losing it all during 5,000 plays).
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