Every game, tested the same way

One board, three readings, two corrections

Per game. The Joe-corrected chi-square read two-sided, the Beta-mixture e-process, and the whole-draw conformal martingale. Each is documented on its own tool; what matters here is that all three are computed the same way for every game, so a row is comparable with every other row.

Two-sided matters. A chi-square far below its expectation is data too good to be true — it is how several scientific frauds were caught — so the p reported is twice the smaller tail.

Across games: Benjamini–Hochberg. Sort the p-values and find the largest $i$ with

$p_{(i)}\le \frac{i\,q}{m}.$

Declare everything up to it. This controls the expected share of declarations that are false rather than the chance of any single one, which is the right thing across hundreds of games: Bonferroni here would need $p<0.05/280$ and a real single-game fault would never surface.

Across games: the world e-value. E-values multiply where p-values cannot, so

$E_{\text{world}}=\prod_{g} E_g$

is a single anytime-valid number for the whole board — it can be looked at after any draw of any game without the looking inflating it. The product runs over independent draws: EuroMillions is one draw sold in nine countries, and counting each national listing would multiply the same evidence in nine times.

A world e-value below one is evidence for the machines. That is worth stating, because "we found nothing" and "we found evidence of fairness" are different claims and only the second is what a small e-value licenses.

The quantile plot puts each game's p against where it would sit if all of them were fair, inside the band that chance allows. The $i$-th of $m$ uniform order statistics is $\mathrm{Beta}(i,\,m{+}1{-}i)$, and its 2.5% and 97.5% points are the band.

What is excluded, and why. A game whose recorded pool does not match the pool it was drawn from produces an enormous statistic that has nothing to do with its machine. Two shapes are detected: a pool that grew, leaving a contiguous block of high numbers far behind for the whole era; and one that shrank, leaving high numbers that simply stop appearing. Six of forty-nine over three hundred draws puts the chance a live number never appears at about one in ten million, so two together is a rule change. Those games are listed separately and kept out of the discoveries, the world e-value and the plot.

The boring answer, stated properly

Most people who look up lottery statistics are asking one question, whether or not they put it in these words: is this rigged?

Every other tool on this site answers it for one game at a time, carefully and at length. This page answers it for all of them at once, and the answer is no — but the interesting part is what kind of "no" the data supports.

There is a weak version: we looked and did not find anything. That is nearly worthless. Enough tests on enough games will fail to find plenty of real things, and a page that reported it would be telling you about the limits of its own attention rather than about lotteries.

The strong version is a shape. If every machine in the world is fair, then every game's fairness p-value is a uniform random number between nought and one, and the collection of them lies along a straight diagonal. Any real bias, anywhere, pulls the low end of that line away from the diagonal — and it does so whether or not anybody guessed in advance which game or which number was affected. The line on this page is straight. That is a positive result, and a stronger one than any single game could give.

The e-value says the same thing in one number. E-values multiply, so the whole board collapses to a single figure that is valid no matter how often you look at it, and that figure is a very small number — which means the evidence points toward fairness rather than merely failing to point away from it.

Now the part that took the longest to get right.

The first time this board ran, its three most extreme games looked spectacular: one had an e-value of 6.8 × 10⁴², which if taken at face value would be the most significant finding in the history of gambling. None of them was real. Texas Cash 5 is recorded as six numbers from thirty-seven, and its 36 and 37 come up at half everybody else's rate because they did not exist for the first part of the era. Colorado Lotto, the same. Massachusetts Megabucks is recorded as six from forty-nine and has not drawn a 46, 47, 48 or 49 since 2023 — its pool shrank and nobody wrote it down.

All three are bookkeeping. The pool changed and the era boundary was never recorded, so the arithmetic is comparing numbers that existed against numbers that did not.

This is the thing that makes a board like this dangerous. A test applied to hundreds of games will find whatever is most broken, and what is most broken is almost never the machine — it is the record of the machine. So those games are detected, listed separately, and kept out of every statistic on the page. What remains is a straight line and a small number, and those are worth more than a headline.