Monte Carlo Estimation of Pi

Features

• Estimates pi from the fraction of points inside the quarter circle, times four
• Reports the sample count, the points inside, the estimate and the relative error
• Lists convergence at powers of ten, so you can see how slowly the accuracy climbs
• Sample count is adjustable up to two million points (about a second)
• The error is relative, so it can be compared directly against the 1/√n scale
• The random source is `crypto.getRandomValues`, consumed without reuse between draws
• Each run differs slightly; click Run again for a fresh batch of points
• Everything runs on your machine with no requests

How to use

1. Set the sample count (a million by default)
2. Read the estimate and relative error in the result block
3. Scroll to the convergence rows to see what each extra power of ten buys
4. Click Run again for a fresh set of random points

FAQ

Why does the estimate always wobble around 3.14 instead of being exact?

That is a property of the method itself, not a defect in the implementation. Monte Carlo error falls as 1/√n — a hundredfold increase in samples only divides the error by ten. A million points typically land within about 0.1%, which is roughly three decimals of stability. For an exact pi, use the digits of pi tool, which returns ten thousand built-in decimals.

What is this tool actually for?

Two things. First, it is the smallest runnable example of Monte Carlo — the idea is understood in a sentence, and it demonstrates the same “estimate a fixed quantity by random sampling” that underlies financial pricing, physical simulation and Bayesian inference. Second, it makes the slowness of convergence visible: in the table, a thousand points and a million points are not far apart, even though the sample count grew a thousandfold — an intuition that formulas alone do not build.

How many samples should I use?

It depends what you want to see. Just the shape of convergence: ten thousand points, instant. An estimate close to 3.14159: a million points takes a couple of hundred milliseconds and usually lands within 0.1%. Two million is the ceiling — beyond it the returns are fixed by the method (the √n term) and the wait grows noticeably.