[Sample] Math, code & small explosions

A circle inside a square #

Imagine throwing points at a square from $-1$ to $1$ on each axis. A point lands inside the unit circle when $x^2+y^2\leq1$.

The ratio of areas gives us an estimate:

A tiny simulation #

function estimatePi(samples = 100_000) {
  let inside = 0;
  for (let i = 0; i < samples; i++) {
    const x = Math.random() * 2 - 1;
    const y = Math.random() * 2 - 1;
    if (x * x + y * y <= 1) inside++;
  }
  return (4 * inside) / samples;
}

Read the result carefully #

More samples usually improve the estimate, but randomness does not promise a perfectly smooth journey. The typical error shrinks on the order of $1/\sqrt{N}$.¹

Keep a record #

Run the function a few times with different sample counts. Note the spread, not just the result closest to the value you expected.

Follow the next question #

What would change if the points were not uniformly distributed? What would you measure to detect that? An experiment becomes useful when it creates a better question.