Roll a fair die again and again and average what you get. The running mean does not wander forever — it is pulled toward the true mean μ = 3.5, and the band of possible error shrinks like 1/√n. This is the theorem that makes every average, poll, and Monte-Carlo estimate mean something. Rendered, not quoted.
SOURCE Jacob Bernoulli, Ars Conjectandi, Pars Quarta (1713) — the first proof of the theorem. Facsimile: archive.org/details/arsconjectandi0000jaco
Draw X₁,X₂,…,Xₙ i.i.d. from a fair six-sided die: each face 1..6 with probability 1/6. Exact moments:
μ = E[X] = 3.5 σ² = Var(X) = 35/12 ≈ 2.9167 (discrete uniform, (6²−1)/12).
The estimator is the sample mean x̄ₙ = (1/n)∑Xᵢ. By linearity E[x̄ₙ] = μ (unbiased) and, by independence, Var(x̄ₙ) = σ²/n — the spread collapses as n grows.
This is the ground the-monte-carlo stands on. Every empirical estimate — a poll, an integral estimated by sampling, a training-loss average over a minibatch — is a sample mean betting on this theorem.
The sample mean is a consistent estimator: x̄ₙ → μ in probability. Averaging works not by faith but because Var(x̄ₙ)=σ²/n → 0. Chebyshev turns that shrinking variance into a hard probability bound.
Live re-check. At boot the panel sealed a known value x̄₂₀₀ and the four LLN invariants. This badge re-runs the verification on demand — if window 6 tampers with the divisor, the sealed value no longer matches and the badge flips red.
A fixed-seed PRNG (mulberry32) emitting i.i.d. fair-die draws. Deterministic — the same seed replays the same stream, so every claim below is reproducible.
Running sample mean vs the true mean line at 3.5, with the ±σ/√n error band. Watch the trace get pinned to the line.
The band is theoretical (σ²/n); the trace is real draws. The trace lives inside the band with high probability — that is the law, live.
Proven at boot over a fixed seed, verified in selfcheck():
The wall also needs finite variance. A Cauchy sample mean never settles — its variance is infinite, so σ²/n is meaningless and the theorem simply does not apply.
“After a run of low rolls, high rolls are due to balance the average.”
→ The gambler’s fallacy. The average converges by dilution — new independent draws swamp early deviations — never by compensation.
“Large n makes the total ∑Xᵢ close to n·μ.”
→ False. The sum’s spread grows like √n and its absolute gap from n·μ diverges. Only the mean converges.
“LLN says x̄ equals μ eventually.”
→ It says x̄→μ in probability; equality has probability zero at every finite n.
Planted void, disclosed. This swaps the divisor from n to n−1 — a biased estimator with E[x̄]=μ·n/(n−1) ≠ μ. The panel value shifts and the sealed boot value no longer matches. The WITNESS (7) catches it live.