Conditional Harmony

Hold every solo frequency still. Do the pairs survive?

A hot ball automatically makes all its pairs look hot. Curveball shuffles the incidence matrix while preserving every draw size and every ball total, then asks what remains.

Largest conditional pair residuals

Texas Two Step4/35 + 1/35 era · since Sep 2025105 draws

3 + 33+3.90σ

6 + 26+3.60σ

3 + 26+3.59σ

7 + 14+3.45σ

25 + 26+3.42σ

17 + 30+3.07σ

12 + 20+3.03σ

28 + 30+2.95σ

19 + 20+2.70σ

7 + 22+2.69σ

11 + 29+2.58σ

1 + 22+2.57σ

9 + 26+2.54σ

1 + 3+2.53σ

19 + 27+2.48σ

29 + 34+2.38σ

20 + 32+2.37σ

2 + 4+2.34σ

Curveball conditional Monte Carlocurveball-pairs-v1seed 2026082759 samples; row and column margins fixed

Observed bars sit over the conditional mean. Ranking by |z| is exploratory across every pair.

The maths, in plain language

Condition on the margins

Represent draws as a binary draw-by-ball matrix. A Curveball trade exchanges non-shared entries between two rows, preserving both row sums $k$ and every column total. Pair counts in the randomised matrices form a conditional reference distribution, read as $z=(O-\mu)/\sigma$ .

The story behind it

The glow a hot ball lends its neighbours

Most “favourite pair” tables ask the wrong question. If 7 appears often, every pair containing 7 inherits the glow. Conditioning removes that easy explanation. The remaining extremes still face a large look-elsewhere effect: this page ranks clues, not discoveries.