T-14 · Probability theory

Chance, counted.

Probabilities are exact fractions where counting allows, and samples where sampling is the point. Every simulation is seeded, so a result can be reproduced and checked against the exact answer. Independent variances add like the squares on the legs of a right triangle, and random error shrinks like one over the square root of the effort.

Rows n = 12

Exactly

Each ball makes n independent left-or-right choices, so the bin it lands in is binomial. Counting paths gives every probability as an exact fraction.

P(S = 6)	231/1024 ≈ 0.225586
Σ P(S = k)	exactly 1
mean np	6
variance np(1 − p)	3

Variances add like squares

Independent bounces are perpendicular in the space of random variables, so their spreads combine by Pythagoras: σ = √(σ₁² + … + σₙ²), which here is √(n p(1 − p)).

σ	√3 = 1.732051
sample mean	6.03 vs 6, standard error 0.0548 within 3 standard errors
sample variance	3.16627 vs 3
TV(sample, exact)	4.4e−2

The central limit theorem

The normal curve approaches the binomial as n grows. The total variation distance measures the largest disagreement about any event; when p = ½ the skew vanishes and the gap falls like 1/n, otherwise like 1/√n.

n	TV(binomial, normal)	ratio
4	1.4e−2	—
8	8.9e−3	1.58
16	4.5e−3	1.971
32	2.5e−3	1.845
64	1.2e−3	2.068

Doubling n divides the gap by roughly 2 (order 1/n); the ratios wobble because the whole numbers k fall in and out of step with the curve.