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 6 numbers contains $\binom{k}{2}$ pairs — every number paired with every other. For Mega-Sena that is 15 pairs in each draw.

The pool of 60 numbers can form $\binom{N}{2} =$ 1,770 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: $D =$ 3,065 draws, so $E =$ 26.0 times for every one of the 1,770 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 1,770 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 1,770 cells at once means running 1,770 simultaneous tests, and chance alone should push about 25 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.

The harmony of numbers

The name of this page is a small bow to Pythagoras, who believed the universe was built out of numerical relationships — that the planets moved at intervals which, if you could hear them, would sound like music. The musica universalis, the music of the spheres.

He was wrong about the spheres and profoundly right about the appeal. Two and a half thousand years later, the instinct that there must be relationships between numbers is still the reason a grid like this one is mesmerising. We do not look at it and see 2,346 independent counts. We look at it and see shapes.

The mirror

That instinct has a name, or several. Apophenia is the general tendency to perceive meaningful connections between unrelated things. The clustering illusion is its specific form here: randomness is lumpy, and lumps read as structure.

Kahneman and Tversky pinned down why in 1972 with the representativeness heuristic — we judge how random something is by whether it looks the way we imagine random ought to look. An even sprinkle looks random to us. Real randomness clumps, streaks and leaves gaps, so genuine randomness reads as suspicious and fabricated evenness reads as fair. It is exactly backwards, and it is close to universal.

The matrix above is not a picture of the lottery. It is a mirror.

How many "special" pairs to expect

There is a formal version of the trap: the look-elsewhere effect. If you search enough places, you will find something remarkable somewhere, and its remarkableness is an artefact of how hard you looked.

Physicists take this so seriously that they will not use the word discovery below five standard deviations — not because five sigma is magic, but because they scan so many places at once that anything less is routinely produced by noise. The banner above the matrix is the same idea, sized for this page: it tells you how many striking cells to expect before you go looking.

When the pattern was real

None of this makes deviation-hunting pointless. It is precisely how cheating gets caught — the difference is knowing the baseline first.

In 2006 Jeffrey Rosenthal was asked to look at Ontario lottery data. Retailers were winning major prizes: around 200 wins, where chance predicted roughly 57. That is not a striking cell in a large matrix. That is a hole in the floor. The investigation that followed ended careers and sent people to prison.

Rosenthal knew what to expect before he looked, which is what made the gap meaningful. Deviation hunting is how statisticians catch cheaters — not how players pick numbers.