Analyzing π through Monte Carlo Simulations

A Comparative Study with Experimental Approaches

Abstract

This study investigates the estimation of π using the Monte Carlo Simulation Method, comparing the results with experimental values. To determine π's experimental value, a unit circle (z=1) centered at the origin within a square bounded by points (0,0), (1,0), (1,1), and (0,1) is considered, where an infinite number of points exist within both the circle and the square. Points with z≤1 are within or on the circle's arc, and those with z>1 are outside the arc but within the square. By selecting hundreds or thousands of random number pairs, their positions relative to the arc and square are determined. With N representing the total number of points considered and n the number of points within or on the arc, the experimental value of π is calculated as π=4n/N. This formula indicates that a larger sample size, N, results in a π value closer to its true value. Furthermore, non-parametric hypothesis testing, such as Friedman's Test, is applied to a Monte Carlo Simulation distribution of 20 triplets of random numbers to evaluate their distribution on a semicircle, followed by a Chi-Square Test for goodness of fit. This comprehensive methodology elucidates various insights into the distribution and impact of random number triplets and their conformity to the expected goodness of fit.