Monte Carlo Estimation of Pi

The procedure for estimating [latex]\pi[/latex] is straightforward and highlights the core Monte Carlo principle. Consider a unit square in the Cartesian plane with vertices at (0,0), (1,0), (1,1), and (0,1). A quarter circle of radius 1 is inscribed within this square, centered at the origin. The area of the square is 1, and the area of the quarter circle is \(\frac{\pi(1)^2}{4} = \frac{\pi}{4}\). The ratio of the quarter circle’s area to the square’s area is therefore \(\frac{\pi}{4}\).

To estimate this ratio, we generate a large number, \(N\), of random points \((x, y)\) where both \(x\) and \(y\) are uniformly distributed between 0 and 1. Each point has an equal chance of landing anywhere within the square. A point \((x, y)\) falls inside the quarter circle if its distance from the origin is less than or equal to 1, which is determined by the condition \(x^2 + y^2 \le 1\). We count the number of points, \(M\), that satisfy this condition. The ratio \(\frac{M}{N}\) is an estimate of the ratio of the areas, \(\frac{\pi}{4}\). Therefore, we can approximate \(\pi\) as \(\pi \approx 4 \frac{M}{N}\). According to the law of large numbers, as \(N\) approaches infinity, this approximation converges to the true value of \(\pi\). However, the convergence is slow, with the error decreasing proportionally to \(\frac{1}{\sqrt{N}}\), making it a very inefficient method for calculating \(\pi\) to high precision compared to deterministic algorithms.