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 $w$.
@article{1176342724,
author = {Berry, Donald A. and Heath, David C. and Sudderth, William D.},
title = {Red-and-Black with Unknown Win Probability},
journal = {Ann. Statist.},
volume = {2},
number = {1},
year = {1974},
pages = { 602-608},
language = {en},
url = {http://dml.mathdoc.fr/item/1176342724}
}
Berry, Donald A.; Heath, David C.; Sudderth, William D. Red-and-Black with Unknown Win Probability. Ann. Statist., Tome 2 (1974) no. 1, pp. 602-608. http://gdmltest.u-ga.fr/item/1176342724/