Inspired by reader Sluggo I am going to take a look at the issue of convergence. I will use the simple Monte Carlo method to estimate pi introduced in my previous post. We are going to briefly explore how the estimate of pi converges on the true value as we increase the size of our trial (the number of random samples we use).
If we know how accuracy varies with the number of trials, we can better design our method to achieve the accuracy we desire without having to iterate; we can better manage the trade-off between accuracy and absorbing computing resources.
I plan to explore the relationship between the accuracy of the estimate of the integral and the fraction of the search space occupied by the integral in a future post.