Math

The average settles, the count does not

A chain you can read - outcome, sum, mean - and two charts on one log-n axis with one cursor: the mean's deviation walks into a funnel of ±2σ/√n, while the excess count S_n − nμ grows like √n and never settles. Predict first whether the excess goes to zero too, then let the run climb; six other runs under the same conditions stand beside it. A forced streak is forgotten by the mean and kept by the count; for the coin the numbers are exact from the binomial; and a Cauchy source, which has no mean, never settles at all.

By Mykhailo Makarov

Actual application image

What happens

Toss a coin a thousand times and the fraction of heads is near one half; toss it a million times and it is nearer. That is the law of large numbers, and it is more precise than it sounds: the fraction converges as 1/√n, and the band the panel draws holds the running mean about 95 % of the time. But the excess of heads over tails does not shrink - it grows, as √n, and a streak of ten heads forced at the start is still there, uncorrected, thousands of tosses later. The law is about fractions, not counts. Toss, watch the mean settle and the pile of excess climb, then force a streak.

Equations used in the module

Equations
running mean m_n = S_n/n → μ, scatter σ/√n
excess S_n − nμ: scatter σ√n - it GROWS
fair coin: E|S_n − n/2| = √(n/2π)
Chebyshev: P(|m_n − μ| ≥ kσ/√n) ≤ 1/k²

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