Imagine playing a game for money in which marbles are drawn out of a bag and then replaced. 60% of the marbles are white. If one of the white marbles is drawn out, you win whatever you risked. The other 40% are blue. If one of the blue marbles is drawn, then you lose whatever you risked. This game has an expectancy of 20¢. That is, over a large number of trials, you’ll make 20 cents for every dollar you risk.

So I wrote a quick program to simulate this game. You start with $1000 and can have up to 100 draws (or until you blow up). Obviously, the simplest approach is to set the bet to a fixed amount and never change it. That is my "system". So I ran a simulation: each run is 10000 games. One game is 100 draws or until the player blows up. The X-axys is the bet in dollars. The red line is the percentage of losses in these 10000 games. The black line is the risk to reward ratio (max gain divided by max loss in these 10000 games).

I have a few questions about the shapes of these graphs. The red one (the percentage of losses) appears to start growing exponentially then it exhibits staircase jumps when the bet reaches a certain value, such as 200,250,500,etc. And it always seems to be a pretty nice round number. Is there a theoretical explanation for these jumps or is it possibly a glitch in my program? Also the stair steps are getting wider after every jump.

And another thing, there seems to be a sweet spot around $100 where the black line hovers over the red one which may mean thats the optimal bet for this game? I also noticed as the bet is approaching $100 and higher, most losses become total blow ups.

Is there a theoretical approach to calculate the optimal bet under these conditions:

- 60% chance to win

- the initial amount is $1000

- up to 100 draws

- the bet is a fixed amount

- the bet ???