You are playing a game where you put in a certain amount of money $m$. A random number in $[0, 1]$ is chosen. If the number is greater than $p$, you now have k% more money, otherwise, you lose all of your money. You can play the game as many times as you want, but you have finite and finite time. What strategy would optimize your return?
Let's say the probability of winning is 0.6.
You have \$$10$. You put \$$1$ in and win 15%. So you have \$$1.15$. You play again with that \$$1.15$, but this time you lose. \$$1.15$ -> \$$0.00$.
But you still have \$$9$ left. You put \$$2$ in and win 15%, (\$$2.30$). You keep it in and win again (\$$2.645$). And you play one more time and win again (\$$3.04165$). You decide to withdraw from the pot, so your total is \$$12.04165$.
Obviously, if you put the whole \$$10$ and play until you lose, your expected value is \$$0$. But the expected value of just one game is positive...