Every pair of numbers, and how often they meet

How often should any two numbers meet?

Take it one step at a time.

A single draw of 5 numbers contains $\binom{k}{2}$ pairs — every number paired with every other. For Hit 5 that is 10 pairs in each draw.

The pool of 42 numbers can form $\binom{N}{2} =$ 861 different pairs in total.

Nothing distinguishes one pair from another — the machine has no idea which balls are which — so by symmetry every pair is equally likely to be among the ones a draw produces. Each draw therefore gives any particular pair a $\binom{k}{2} / \binom{N}{2}$ chance of turning up, and across $D$ draws:

$$E = D \cdot \frac{\binom{k}{2}}{\binom{N}{2}} = D \cdot \frac{k(k-1)}{N(N-1)}$$

For this era: $D =$ 2,230 draws, so $E =$ 25.9 times for every one of the 861 pairs. That is what the neutral colour on the matrix means. Warm is above it, cool is below.

For the classic 6/49 game the same arithmetic gives $D \cdot 15/1176 \approx 0.01276 \cdot D$ — about 13 times per pair over a thousand draws.

Reading a single cell

Co-occurrence counts are small and cannot go below zero, so the natural scale for their spread is the Poisson one:

$$z = \frac{O - E}{\sqrt{E}}$$

A cell two standard errors from the middle looks striking. On this matrix, it should not.

The whole matrix in one honest picture

Here is every one of the 861 z-values, against the bell curve they would follow if nothing whatsoever were going on:

The bars are this lottery. The gold curve is chance. Any individual cell you could point at is one bar's worth of that shape.

Why some cells always look special

This is the trap, and it is arithmetic rather than opinion. Showing 861 cells at once means running 861 simultaneous tests, and chance alone should push about 12 of them past $|z| > 2.5$. The banner above the matrix counts the real ones live, beside that figure.

With 50 numbers — EuroMillions — there are $\binom{50}{2} = 1{,}225$ pairs, so at the conventional $\alpha = 0.01$ about twelve will look "significant" in any fair history you care to generate. Finding a striking pair in a matrix this size is not a discovery. It is the expected outcome.

One technical note, because it changes the number above by half. These counts are small — around 26 per pair — and at that size the familiar normal-curve tail is a poor fit for a discrete, non-negative count. Using it here would have claimed 29 extreme cells for Powerball where a fair machine actually produces about 19, which would have made the real lottery look quieter than chance. The figure above comes from the exact Poisson tail instead, and is checked against simulated fair histories in the test suite.

That is why the honest thing to do is press shuffle reality and watch a matrix built by a random number generator produce exactly the same kind of structure.

1. Bonferroni, C. E. (1936). Teoria statistica delle classi e calcolo delle probabilità.