Study of the Value of π Probability Sampling by Testing Hypothesis and Experimentally

Abstract

This study evaluated the value of π using the Monte Carlo Simulation Method and compared the results with experimental values. The experimental value of π was determined by considering a unit circle |z| = 1 centered at the origin, inscribed within a square with vertices (0, 0), (1, 0), (1, 1), and (0, 1). Points were randomly generated within the square, where points satisfying |z| ≤ 1 lay within the circle, and those with |z| ≥ 1 lay outside the circle but within the square. By selecting large numbers of random pairs and determining their positions relative to the circle, the ratio π = 4n/N was calculated, where N was the total number of points and n was the number of points within the circle. Larger sample sizes yielded values of π closer to the true value. The distribution of Monte Carlo Simulation results, using 20 triplets of random numbers, was examined with non-parametric tests such as Friedman’s Test. Ranks were assigned to the 20 random numbers row-wise for each triplet. The null hypothesis, asserting that all triplets had identical effects, was tested and showed significant differences at the 5% level. Additionally, the distribution was tested for goodness of fit using a Chi-Square Test at a 5% significance level. Results indicated that the triplets of random numbers conformed to the expected distribution.