The Limit Shape
Every Powerball white-ball draw is a sorted 5-tuple from 1–69. Take any single draw and it looks like nothing — five integers scattered across the range. Stack 1,300+ of them and a deterministic shape emerges: five overlapping humps, one for each order statistic. We compare what the lottery actually drew to the closed-form order-statistic distribution, then watch the empirical shape converge to the theoretical curve as N grows.
| Order stat | Theoretical μ | Real μ | Sim μ | KS (real) | KS (sim) |
|---|---|---|---|---|---|
| 1st | 11.667 | 11.934 +0.268 | 11.564 | 1.98% | 1.53% |
| 2nd | 23.333 | 23.632 +0.299 | 22.934 | 2.40% | 2.48% |
| 3rd | 35.000 | 35.604 +0.604 | 34.539 | 3.12% | 3.07% |
| 4th | 46.667 | 47.151 +0.485 | 46.145 | 2.83% | 3.18% |
| 5th | 58.333 | 58.595 +0.262 | 58.298 | 2.29% | 2.21% |
KS distance is the largest gap between the empirical CDF and the theoretical CDF for that order statistic. Real and Sim rows are matched in size — comparing them shows how much of the “wiggle” in the real data is just sampling noise.
How it works
The encoding. For each draw, sort the five whites ascending: x₁ < x₂ < x₃ < x₄ < x₅. The k-th value is the k-th order statistic. Across all draws, the empirical distribution of xₖ for each k = 1..5 is what we plot.
The closed form. Under uniform sampling without replacement, the k-th order statistic of a 5-subset of {1..69} has the exact PMF P(xₖ = v) = C(v−1, k−1) · C(69−v, 5−k) / C(69, 5). That gives five known curves tiling the range, with peaks near v ≈ k · 70/6 ≈ 11.67, 23.33, 35, 46.67, 58.33.
The noise envelope. How close to perfect should a finite sample look? We answer that with simulation: at each milestone size (50, 100, 200, …, all draws), we draw 50 independent batches of fair uniform samples, compute the L∞ distance between each sample's empirical PMF and the theoretical PMF, and take the 95th percentile. That's the noise envelope — the line a fair lottery's deviation should sit under 95% of the time.
What it tests. Inspired by Igor Pak's 2003 paper on the asymptotic stability of partition bijections [pdf], which formalizes the idea that random combinatorial objects have a deterministic limit shape. Sylvester saw this in the 1880s for sorted partitions; we see it here in 1,300+ Powerball draws. If the lottery is fair, the empirical shape should converge to the theoretical at rate 1/√N, and stay inside the envelope.