Three running means of a die, wild for the first rolls and narrowing into the line at 3.5 as n grows, with the ±0.5 band they enter and stay in — the single rolls never settle, their average does. The Central Limit Theorem's pair: that one says what shape the means take, this one says where they go. Evidence: how often a run is still outside the band at each n; the ladder is the rolls it takes to pin the mean to a tolerance.
Heads catches up
Flip a coin ten times and you might get seven heads — that looks unfair. Keep flipping all afternoon and count: heads and tails end up almost exactly even. The coin never remembers what it did; it is just that one lucky streak gets buried under thousands of ordinary flips.
Streaks are real, but they get drowned — that's the whole law.
x̄ₙ = Σ Xᵢ / n · P(|x̄ₙ − μ| > ε) → 0 as n → ∞
Within ±1
a rough guess
n ≈ 12
Within ±0.5
the band above
n ≈ 45
Within ±0.1
one decimal
n ≈ 1,121
Within ±0.01
two decimals
n ≈ 112,047
Bars are log₁₀ n — the rolls needed for 95% of runs to sit within the tolerance of 3.5; ten times the precision costs a hundred times the rolls