🎯 Monte Carlo Pi Estimation

A quarter circle of radius 1 inscribed in a unit square has area π/4, while the square has area 1. If points are scattered uniformly across the square, the probability any one lands inside the quarter circle equals that area ratio, π/4. Counting the fraction of points that land inside and multiplying by 4 therefore estimates π.

Each batch of random points is a fresh statistical sample, so the observed inside-fraction fluctuates around the true value of π/4. As the total number of points N grows, the law of large numbers pulls the running estimate closer to π, and the typical error shrinks proportionally to 1/√N.

It is a statistical estimate rather than an exact calculation. Because each point's position is random, the count of points inside the quarter circle is itself a random variable, so the estimate always carries some sampling error. That error shrinks with more samples but never fully disappears in a finite run.

Monte Carlo estimates typically converge at a rate of 1/√N, meaning you need roughly 100 times more samples to gain one extra decimal digit of accuracy. This is much slower than deterministic numerical methods for smooth one-dimensional problems, but Monte Carlo scales far better to high-dimensional problems where those methods become impractical.

Monte Carlo methods are used wherever a problem is too complex to solve exactly but can be approximated through repeated random sampling: pricing financial options, simulating particle physics and nuclear reactions, rendering global illumination in computer graphics, optimisation, and Bayesian statistical inference, among many other applications.