Every number, and how many people hold it

Counting the crowd directly

In a ball game the crowd has to be inferred: a prize table says how many tickets matched $m$ of the numbers drawn, and untangling which numbers those tickets held takes elementary symmetric polynomials and a lot of care. A digit game hands the answer over. The straight tier is won by holding the exact number drawn, so its winner count is the number of people on that number.

The model. For a draw $d$ whose number was $x_d$, in a game of $m$ digits,

$W_d \sim \operatorname{Poisson}\!\left(S_{c(d)}\;\pi(x_d)\;10^{-m}\right),$

where $S_c$ is the straight tickets sold, held constant inside a cell $c$ of one draw slot and one calendar month, and $\pi$ is popularity with mean one over the $10^m$ numbers. The level of $\pi$ is not identified separately from , which is why is normalised and carries the scale.

Profiling out sales. For fixed $\pi$, the maximum-likelihood level of a cell is a ratio of sums:

$\hat S_c=\frac{\sum_{d\in c} W_d}{\sum_{d\in c}\pi(x_d)\,10^{-m}}.$

So the sales nuisance never enters the search. Each round recomputes it in closed form and the optimiser only ever moves the popularity parameters.

What popularity is made of.

$\log\pi(x)=\sum_{j=1}^{m}\alpha_{j,x_j}\;+\;\beta_{\text{rep}(x)}\;+\;\gamma\,\mathbb{1}[x\text{ reads as a date}]\;+\;u_x .$

The first term is a free weight for each digit in each position, with digit zero held out; the second is one indicator for the deepest repeat the number contains; the third marks numbers a person could reach by writing down a birthday. The named block is fitted by Newton steps on its full information matrix — the columns overlap heavily and treating them one at a time diverges.

The residual, and why it is shrunk. $u_x$ is a free weight for the number itself, penalised by $\lambda\sum_x u_x^2$. Its per-number Newton step is one-dimensional given everything else, so the whole block costs a single pass. The penalty is not decoration: a four-digit game has ten thousand numbers and perhaps six thousand draws to fit them with, so most numbers are seen once or never. is therefore chosen on a 70/30 chronological holdout, with "no residual at all" in the running — and on games where the digits are the whole story, that is what wins.

Honesty. Dispersion runs between about 2 and 60 rather than 1, and every standard error is widened by its square root. The map is published only when it beats knowing the month's sales alone on draws the fit never saw.

What the model cannot see. Popularity is constant in time here. When a number means something on one particular day — a date, an anniversary — the count exceeds anything $\pi$ can produce, and those draws appear as surges rather than being absorbed.

The numbers people cannot help playing

A three-digit lottery has a thousand numbers in it, and they are not played anything like equally.

In Connecticut's Play 3, a number made of three identical digits — 111, 222, 777 — is held by twenty to thirty times as many tickets as an ordinary one. In Delaware the same numbers run to forty and fifty times. Four digits are stranger still: four of a kind is held by fifty to eight hundred times the average, depending on the state, and the effect is so large that a single draw of 7777 in Connecticut paid 2,067 straight tickets on a night when the typical number pays five.

The rest of the map is quieter and more human. In every Pick 4 in this data the leading digit is far more likely to be 1 than anything else, and the digit after it leans toward 9. That is a pool of people playing the year somebody was born. In Pick 3 the middle digit runs from about 2.5 times average at 1 down to 0.4 times at 9, which is the same instinct in a smaller space: people write down small numbers and dates, and 9 is neither.

And some numbers mean something on one particular day. Connecticut drew 911 on 11 September 2021, the twentieth anniversary, and 8,122 tickets held it — the largest straight count in that game's recorded history, and nearly twice what even that number's own considerable popularity predicts. It came out of a search for corrupted rows. Across every game here, a number drawn on the day it spells is held about 1.4 times as often as usual, and that figure comes from twelve games that had no part in noticing the pattern.

Here is what makes this worth a page rather than a curiosity. In almost all of these games it costs the player nothing. A fixed-prize Pick 3 pays $250 to every straight winner whether five people held the number or five thousand. The crowd is visible and completely harmless.

The exception is the game Herman Chernoff was looking at. In a pari-mutuel daily numbers game the winners share a pool, so a popular number really does pay less — and Chernoff's insight in the early 1980s was that the unpopularity of some numbers was large enough to be worth acting on. Of the games mapped here only California's Daily 3 measurably works that way, and the page says so before you reach the map. Everywhere else this is a portrait of what people choose, and nothing more.