A gambler seeks to maximize his probability of reaching a goal in a game where he is allowed at each stage to stake any amount of his current fortune. He wins each bet with a certain fixed probability $w$. Lester E. Dubins and Leonard J. Savage found optimal strategies for a gambler who knows $w$. Here strategies are found which are uniformly nearly optimal for all $w$ and, therefore, also for a gambler with an unknown $w$.
"Red-and-Black with Unknown Win Probability." Ann. Statist. 2 (3) 602 - 608, May, 1974. https://doi.org/10.1214/aos/1176342724