The martingale strategy has also been applied to roulette, as the probability of hitting either red or black is close 🎅 to 50%.
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 expected amount lost is (63 × 0.021256)= 1.339118.
In a unique circumstance, this strategy can make sense. Suppose the gambler 🎅 possesses exactly 63 units but desperately needs a total of 64. Assuming q > 1/2 (it is a real casino) 🎅 and he may only place bets at even odds, his best strategy is bold play: 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.[2] 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.[3]
Alternative mathematical analysis [ edit ]
