Average Error as Number of Samples Increases

I made a very simple program that approximates $$\pi$$ in r, by finding the probability that 2 random generated numbers are coprime for n trials. The result of this probability approaches $$\frac{6}{\pi^2}$$ wiht more trials, so from there you can isolate $$\pi$$ and return an approximation of $$\pi$$. For more info, refer to this page: https://en.wikipedia.org/wiki/Coprime_integers. From this it is obvious that as your trials n approaches infinity, the algorithm would return the exact value of $$\pi$$. Hence I made the program return the percent error between the approximation and the true value of $$\pi$$. I began to wonder what would happen if you found the average of m percent errors for a set n. Would this average error converge to a specific percent error as your m approaches infinity, and would it be related to n? I am both a beginner mathematician and a programmer, so I would not know how to approach this problem theoretically or practically.