What happened next to everything that looked remarkable

Replaying an era, and reading the afterlife

The replay is causal. Walk the era forward in strides of ten draws, and at each point compute only what was knowable then. At draw $t$, a number's reading is

$z=\frac{O_t-tp}{\sqrt{tp(1-p)}},\qquad p=\frac{k}{N},$

a pair's is the Poisson $z$ against $t\,k(k-1)/N(N-1)$, and a wait's is the current gap against the geometric mean $1/p-1$ with standard deviation $\sqrt{1-}$.

A crossing is the first time a cell passes the threshold its own surface would have used — the two-sided Bonferroni $z$ for that surface's cell count, so $N$ for numbers and waits and $\binom{N}{2}$ for pairs. On a 6-from-49 game that is 3.29 for a number and 4.26 for a pair. Each subject is recorded once: the ledger is a list of moments at which this site, had it existed then, would have shown a visitor something remarkable.

The afterlife is measured on fresh draws. For a crossing at index $c$ and a horizon $h$, the later reading uses only draws $c$ to $c+h$:

$z_h=\frac{O_{[c,\,c+h)}-hp}{\sqrt{hp(1-p)}}.$

This is the part that matters, and getting it wrong would invalidate the whole page. A cumulative count carries the original excess forward for ever — a number that ran 30 ahead of expectation stays 30 ahead unless something actively pulls it back — so a cumulative reading would show a decay that is pure arithmetic and says nothing about the game. Fresh counts are out of sample. An anomaly that was real stays high; one that was luck returns to nothing, and is free to land on the other side of zero.

A wait's afterlife is read as a count, not as another wait. Once a drought has ended it has nothing more to say; the question that survives is whether the number went on being rare.

Nothing to correct for afterwards. The crossings were selected by a corrected threshold, and the horizons are then measured on draws that took no part in that selection. The selection is the multiplicity, and it has already been paid for.

Why the mean and not the row. One crossing's afterlife is a single $z$ with a standard deviation near one, so even a genuinely biased ball can read low on a given horizon — measured, a 35% lean predicts about 1.8 at 200 draws and lands anywhere from 0.5 to 3 depending on where the era happened to start. The average over many crossings is the figure with something to say, and it is stated before the table for that reason.

Marking our own homework

Every statistics page on the internet has the same missing paragraph. It tells you that something is unusual — this number is overdue, this pair comes up together more than it should, this drought is the longest on record — and then it stops. It never comes back a year later to say what happened.

This page is that paragraph.

The method is simple enough to describe in a sentence: replay a lottery's history from the beginning, and every time some number, pair or waiting-time crosses the threshold that this site's own tools would have used, write it down. Then look at what those same things did over the next fifty, hundred and two hundred draws — using only the draws that came after, so it is fresh evidence and not the same excess being counted twice.

On the UK's Lotto, thirty-four things crossed. Their average reading at the moment of crossing was 3.9, comfortably past the corrected threshold, the kind of number that gets a chart drawn around it. Over the two hundred draws that followed, those same thirty-four averaged 0.07.

Powerball: forty-two crossings, average 3.8 at the time, 0.07 afterwards. SuperEnalotto: sixty-one crossings, 3.7 at the time, −0.12 afterwards.

That is regression to the mean, and there is nothing surprising about it — which is exactly why it is worth showing. When you examine forty-nine numbers, and eleven hundred pairs, and forty-nine waiting times, at three hundred points in a game's history, some of them will cross any threshold you set. That is not a flaw in the threshold. It is what a threshold is: a promise about how often you will be wrong, kept.

The honest version of every "hot number" story is on this page, in the fourth column.

One caution, and it applies to reading this page as much as to reading any other. Do not draw a conclusion from a single row. One crossing's afterlife is one number, with a standard deviation near one, so an anomaly that was genuinely real can still read low on any particular horizon — the tests behind this page show a truly biased ball reading anywhere between 0.5 and 3 depending on nothing more than where its era happened to start. It is the average over many crossings that carries the weight, and it is printed first for that reason.

The thing this page cannot do is tell you which of its rows was real. If a lottery here did have a heavy ball, it would appear as a crossing whose afterlife stayed high — and one such row, on its own, would look much like a lucky one. What separates them is more draws, which is the answer to almost every question on this site.