There is one real edge.
It is not the one you think.

One identity, doing all the work

A jackpot is pari-mutuel: the prize is fixed, and everyone holding the winning combination divides it. So if $X$ other people hold your numbers, you receive

$$\frac{1}{1 + X}$$

of the prize. $X$ is a random variable — you cannot know it when you buy — so what you can reason about is its average.

Model the other tickets as scattered independently across the $\binom{N}{k}$ combinations. Then the count holding yours is Poisson with mean

$$\lambda = (S - 1)\,\frac{m}{\binom{N}{k}}$$

where $S$ is the number of tickets sold and $m$ is how many times likelier your particular combination is to be chosen than an indifferent picker would choose it. For a ticket nobody favours, $m = 1$.

Now the expectation. It looks as though it should need numerical work, and it does not:

$$\mathbb{E}\!\left[\frac{1}{1+X}\right] = \sum_{j \ge 0} \frac{1}{1+j}\, e^{-\lambda} \frac{\lambda^{j}}{j!} = \frac{e^{-\lambda}}{\lambda} \sum_{j \ge 0} \frac{\lambda^{j+1}}{(j+1)!} = \frac{e^{-\lambda}}{\lambda}\left(e^{\lambda} - 1\right) = \frac{1 - e^{-\lambda}}{\lambda}$$

The trick is in the middle step: $\frac{1}{1+j} \cdot \frac{\lambda^j}{j!}$ is , which re-indexes the sum into the exponential series with its first term missing.

That closed form is the entire feature. Everything this page claims about unpopular numbers is this one expression evaluated at two different $\lambda$.

Two limits are worth holding onto. As $\lambda \to 0$ the share tends to 1 — if nobody else is likely to hold your numbers, you keep the lot. And for large $\lambda$ the share behaves like $1/\lambda$, so once a combination is genuinely crowded, doubling the crowd halves your slice. The interesting region is between them, which is exactly where real lotteries sit.

What $m$ is, and what it is not

$m$ is a model output, not a measurement. Lotteries do not publish the combinations their players chose, so nobody outside an operator can compute it directly. What the literature does establish, repeatedly and consistently, is the direction of the effects: dates are over-played, 7 is over-played, numbers above 31 are under-played, tidy-looking tickets are over-played.

The parameter table on this page is a set of multiplicative weights $w_n$ chosen to reproduce those directions, and a ticket's $m$ is

$$m = \frac{\prod_{n \in T} w_n}{e_k(w)} \binom{N}{k}$$

where $e_k$ is the elementary symmetric polynomial of degree $k$ — the sum over all $\binom{N}{k}$ combinations of the products of their weights. It is the normaliser that makes the weighted pick distribution sum to one, and it is computed by a dynamic programme rather than by enumeration, which would be hopeless at .

The magnitudes are calibrated so that a typical popular ticket and a typical unpopular one differ in expected share by about the factor of two that Ziemba's 6/49 work reports. They are not precise, and the page says so wherever a number derived from them appears.

Why the edge cannot become a profit

The tempting next thought is that a large enough jackpot must eventually make the bet favourable. It does not, for a reason with three separate parts.

First, the advertised jackpot is not money. On annuity games the headline is the sum of payments spread over decades; taken now it is worth roughly half. Then it is taxed. A \100$m billboard is nearer$$31$m in hand before anyone else is considered.

Second, sales grow with the prize. Ziemba's observation is that a bigger jackpot draws more tickets, roughly as a power law, and more tickets mean a larger $\lambda$ for every combination. The extra sharing eats most of the extra money: the jackpot contribution to expected value is approximately

$$\frac{1}{\binom{N}{k}} \cdot J \cdot \frac{1 - e^{-\lambda}}{\lambda} \;\approx\; \frac{J}{S \, m} \quad \text{for large } \lambda$$

and since $S$ grows with $J$, that ratio flattens out instead of climbing.

Third, the operator's take is deducted before any of this. Roughly half of every ticket sold never enters the prize pool at all.

Put together, the arithmetic on this page returns a loss at every jackpot a real game reaches. The unpopularity effect is real and it is worth something — but what it is worth is a smaller loss.

The gambler's ruin footnote

Suppose the edge were positive. Chernoff's own conclusion, having found exactly that in the Massachusetts Numbers Game, is the one this page ends on.

A bet with positive expectation and enormous variance, played from a finite bankroll, is not a way to make money. The expectation is an average over infinitely many plays; ruin is an absorbing state reachable in a few hundred. The simulation at the foot of the page is that statement made concrete — five hundred players, a genuinely favourable bet, and most of them broke.

The one honest edge, and what it is not

Everything else on this site is an argument that the patterns people see in past draws are not there. This page is the exception, and it is worth being careful about why.

There is a real, documented, measurable effect in lottery play. It has been found repeatedly, by economists using data operators do publish — the number of winners at each prize tier — and it is not controversial. It is this: people do not pick numbers uniformly, and the prizes are shared, so what you pick changes how much you get when you win.

It does not change whether you win. Nothing does. The machine has no idea what anyone chose, and every combination arrives with exactly the same probability it had before anybody bought a ticket.

That distinction is the whole page, and it is easy to lose. "Unpopular numbers give you an edge" is true in a specific, narrow, arithmetic sense and false in the sense a reader is most likely to hear it.

How anyone knows

The obvious objection is that lotteries do not publish which combinations people chose — so how could anyone possibly know that 7 is over-played?

The answer is a nice piece of inference. Operators do publish how many winners there were in each tier. If picks were uniform, the number of winners sharing a prize would follow a predictable distribution given the sales. It does not. Some draws produce far more co-winners than uniform play could explain, and others far fewer, and the pattern of which is which lines up with the numbers drawn.

A draw of 3, 14, 22, 25, 29, 31 produces a crowd. A draw of 34, 38, 41, 44, 47, 49 produces almost nobody. From enough draws you can work backwards to a picture of the preferences that must be behind it.

Cook and Clotfelter did this in 1993. Ziemba and colleagues did it for the Canadian 6/49 and reported that a portfolio of unpopular numbers returned markedly more per dollar than a popular one. Baker and McHale did it for the UK. The direction of the effect is one of the more replicated findings in the economics of gambling.

The size of it is a different matter, and worth being careful about. How much more of a jackpot an unpopular ticket keeps depends on how many tickets are in play that night — at low sales almost nobody shares anything and the advantage nearly vanishes; at high sales it grows. The machine above is a model with the directions taken from that research and the magnitudes chosen by us, and it is labelled as one wherever it produces a number. Treat its percentages as illustrative of a shape, not as measurements.

The tidy tickets

The strongest single effect is the calendar. Millions of tickets are birthdays, anniversaries and children's ages, which is why the pool below 31 is crowded and the numbers above it are not. In a 6/49 game that is more than a third of the pool that most players effectively never reach.

The second is shape. People pick tickets that look designed: runs, even spacing, patterns that make a line on the ticket slip. When the Ontario draw of April 2008 came up 40-41-42-43-44-45, the second prize was shared by 239 people — a tier that usually has a handful of winners. Nobody had predicted the draw. A great many people had independently decided that a neat sequence was worth a pound, and they were all right at once.

Rosencrantz, Guildenstern and the syndicate

The stories where somebody genuinely beat a lottery are all in the case files, and none of them involves this effect. They involve a rigged random number generator, a compromised drawing machine, an insider at the printing works, and one man who simply bought every combination when the prize exceeded the cost of doing so.

The unpopular-numbers effect has never made anybody rich, and the reason is arithmetic rather than luck. The edge shrinks a loss; it does not create a gain. Even where a genuine positive edge has existed — Chernoff found one, and said so in print — the variance is so large that a finite bankroll is far more likely to be gone before the average arrives.

Herman Chernoff's paper on the Massachusetts Numbers Game is unusual and worth reading for its tone alone. He found the edge, described it precisely, and then explained at length why he was not going to bet on it. That is the register this page is trying for.

What to do with this

Nothing, is the honest answer, unless you were going to buy a ticket anyway.

If you were, then picking numbers nobody else picks costs nothing and means that in the vanishingly unlikely event you win, you keep more of it. That is the entire practical content of one of the most-cited findings in the field.

And if the effect has made you feel that a lottery is more beatable than you thought, the page has misfired. It is exactly as unbeatable as it was. The only thing that changed is who you would be splitting it with.