Monte Carlo Explorer — interactive explainer

Samples per run

● inside circle ✕ outside circle

Total samples

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Inside circle

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Estimate of π/4

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Estimate of π

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True π/4

0.78539816339744…

True π

3.14159265358979…

Each "dart" lands at a uniform random point in the square. The fraction landing inside the inscribed circle estimates π/4. Run many experiments to watch 95% confidence intervals accumulate — nearly all should capture the dashed true value.

Experiment History — 95% Confidence Intervals for π/4

Why does dartboard sampling estimate π?

Area of square:(2r)² = 4r²

Area of circle:πr²

Pr(dart inside):πr² / 4r² = π/4

By LLN as N → ∞:(# inside) / N → π/4

Therefore: π ≈ 4 × (# inside circle) / (# total samples)

Borel’s Law of Large Numbers guarantees convergence — the 95% CI will capture π ≈ 95% of the time.

Function Samples

Random rectangles of width L/M and height f(x_i)

Samples (M)

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Domain length (L)

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Rect. width (L/M)

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MC Estimate

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True Integral

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Relative Error

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① Connection: this function's integral equals π/4 — the same quantity estimated by dartboard sampling on Tab ①. Both are Monte Carlo estimates of the same number!

Experiment History — 95% Confidence Intervals for ∫f(x)dx

Random Riemann Rectangles — as M grows, width L/M shrinks

Each rectangle has width L/M and height f(x_i) at a random x_i ~ U(a, b). As M → ∞, overlaps and gaps average out and the estimate converges to the true integral. The formula (L/M)·Σf(x_i) is a Riemann sum with random partition points. This animation always uses f(x) = sin(x) on [0, π].

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