On optimal strategies for a betting game (Q809914)

A one-person game in which the player can bet up to n times is considered. The amount of each bet is between 0 and 1 and the probability to win any bet is \(p\in (0,1)\). The player wins the game if the final result is a given positive integer m. If \(p\leq 1/2\), then the strategy N(n,m) of betting 1 every time until m is reached is optimal. If \(p>1/2\), this strategy is not optimal if n is sufficiently large.