I have created a program that does 1000 simulations of randomly selecting N number of balls (with different colors) from let's say 20 balls. This program also counts how many instances that a particular desired outcome/event (e.g. all N balls are the same color, etc.) has occurred from all those simulations. I may also tweak the program to do 2000, 3000,...,10,000 simulations, and compare the number of instances that the desired event has occurred. Now, my problem is how can I use those figures in a graph to prove that the probability of that event is equal to or approximately equal to X%. I have no problem with making the program itself, but any suggestions for the graph (i.e. x versus y axis) would be appreciated.
Here's an example I put together, simulating a basic fair coin toss. On the X-axis, the number of simulations is shown and on the Y-axis the success rate (e.g. the proportion of times we got heads), which should approximate the probability. (The plot was done using R, and the number of simulations is from 100 to 10000.)
From the picture, you can see that there is, of course, some variance in the data as the experiment is random. However, as we make more simulations, the variance of the data is getting smaller. Also, this plot is a close-up view, and the values on the Y-axis are mostly $0.5\pm0.02$ (the error is smaller near the end), so we have a good approximation. On the next plot, I rescaled the Y-axis so it shows values from $0$ to $1$. The limit can there be easily seen, and is, of course, $0.5$.