Monte Carlo Pi
Source inspiration: (Mathew 2000-2019).
Description
Monte Carlo estimation of \(\pi\) samples random points uniformly in the unit square \([0,1]\times[0,1]\) and counts how many lie inside the quarter unit disk \(x^2+y^2\le 1\). If \(n\) points are sampled and \(m\) of them satisfy the disk test, then \[ \hat\pi_n = 4\frac{m}{n}. \]
The stochastic error decreases on the order of \(1/\sqrt{n}\), so convergence is slower than high-order deterministic quadrature but robust in higher-dimensional settings.
Animations
Each animation below shows the Monte Carlo point-cloud estimator for \(\pi\) in the quarter-disk geometry. Frames follow the worked-example sample sizes \(n=100, 400, 1600, 6400, 10000\).
Julia source scripts that generated these animations are linked under each case.
Case 1 — Monte Carlo \(\pi\) Estimation, Quarter Unit Disk in \([0,1]^2\)
Behavior: Green points fall inside \(x^2+y^2\le 1\) and orange points fall outside. The estimate \(\hat\pi_n\) fluctuates but trends toward \(\pi\) as \(n\) increases.
No archived animation GIF is available for this topic in the current snapshot, so this animation is reconstructed from the module examples.