Abstract
In discrete red and black, you can stake any amount s in your possession, but the value of s takes positive integer value. Suppose your goal is N and your current fortune is f, with 0$\frac{1}{2}$ where the house has the advantage over the player, and with the value of p greater than $\frac{1}{2}$ where the player has the advantage over the house. The optimum strategy at any f when p<$\frac{1}{2}$ is to play boldly, which is to bet as much as you can. The optimum strategy when p>$\frac{1}{2}$ is to bet 1 all the time.