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.
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