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Reading the draws and working out what chance predicts.
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Reading the draws and working out what chance predicts.
The unpopular numbers edge
Every other page here argues that the patterns people see in past draws are not there. This one is the exception: a genuine, documented, repeatedly measured effect in lottery play. It is worth real money, and it has never once made anybody rich — and the gap between those two facts is the whole page.
Nothing on this page changes your chance of winning. It only changes how many strangers you’d share with IF you win.
The map of the crowd
Nobody publishes which combinations players chose. But operators publish how many winners shared each prize, and from enough draws you can work backwards to the preferences that must be behind it. This is what that inference finds — and the cliff in the middle is the calendar.
The cliff at 31 is the calendar. Millions of tickets are birthdays and anniversaries, so every number a date can reach is played more often than the ones above it — and the pool does not stop at 31.
Illustrative model, calibrated to published studies — not live data.
Two tickets, one machine
These two tickets have exactly the same chance of winning: 1 : 11,238,513. Drag the sales up and watch what happens to the money. At 30 million tickets the crowded one keeps 8% of the prize and the sparse one keeps 62%.
A very ordinary ticket
A ticket almost nobody plays
Same odds. The unpopular ticket keeps 8.3× as much of the prize — $11.0m more, on the occasions it wins at all. Pull the sales down and the gap closes: when hardly anyone is playing, hardly anyone shares, and it stops mattering what you picked.
If the other players’ tickets are scattered independently, the number of people holding yours is Poisson with mean λ, and your share of a pari-mutuel jackpot is 1/(1+X). The average of that has a closed form — it is the whole feature in one line:
E[1/(1+X)] = (1 − e−λ)/λ
At λ = 13.29 that is 8%; at λ = 1.03 it is 62%. Nothing else on this page is doing any work.
Nothing on this page changes your chance of winning. It only changes how many strangers you’d share with IF you win.
Illustrative model, calibrated to published studies — not live data.
Your own numbers
Tap in the ticket you actually play. This tells you where it sits and why. It will not suggest a different one — a page that hands out numbers is a betting system with a statistics section attached, and that is the thing the rest of this site exists to argue against.
Tap the 5 numbers you actually play, or take one of the lines you keep. Nothing is stored here, and nothing suggests different ones.
The whole ticket, taken apart
Here is where the edge stops being exciting. The advertised jackpot is an annuity, so taking it now is worth about half; what survives tax is smaller again; then it is shared. Every version of this arithmetic is a loss, at every jackpot a real game reaches. Unpopular numbers make the loss smaller. They never make it a gain.
| Outcome | Chance | Worth per ticket |
|---|---|---|
| jackpot, after cash value, tax and sharing | 1 in 292,201,338 | $0.02 |
| all lower tiers combined | 1 in 35,120 | $0.56 |
| Everything the ticket is worth | $0.58 | |
| What it costs | −$2.00 | |
| Net | −$1.42 |
You get back about $0.58 of every $2.00 — a loss of 71%. Picking unpopular numbers makes that loss smaller. It never makes it a gain.
Nothing on this page changes your chance of winning. It only changes how many strangers you’d share with IF you win.
Illustrative model — jackpot, sales and prize structure are set by these sliders, not read from a live game.
How much of your money the staking rule says to risk
The Kelly criterion is the standard answer to “how much should I stake”, and on a favourable bet it is famously aggressive — it will put a fifth of everything on a good enough edge. That is exactly why it belongs here: nobody can accuse it of timidity, and it still says no.
f* = (b·p − q) / b
f* = -0.000032%
At $20M on Powerball Double Play, one line at $2.00 wins once in 292,201,338 and pays 3,088,385 to one after the cash value, the tax and the sharing. The rule returns -0.000032% of a bankroll — a negative fraction, which does not mean “stake a little”. It means take the other side of the bet, and there is no other side of a lottery to take.
The cheque here is the advertised headline after the three reductions that always apply: the cash option, the withholding, and the share held with everyone who picked the same line. Take any of them out and the fraction is still negative.
What “somebody has to win” actually costs
Distinct lines needed to reach a chosen chance of the jackpot, from n = ln(1 − c) / ln(1 − p). Distinct is the best case: buying the same line twice buys the same chance twice.
| Chance of the jackpot | Distinct lines | Cost | Of every combination |
|---|---|---|---|
| 1% | 2,936,722 | $5,873,443 | 1.0% |
| 10% | 30,786,484 | $61.6M | 10.5% |
| 50% | 202,538,533 | $405M | 69.3% |
| 90% | 672,818,444 | $1.35B | 230.3% |
An even chance costs more than any jackpot this game has ever advertised, which is the arithmetic behind every syndicate that has tried to buy out a draw — and the reason the two that succeeded needed a jackpot several times the cost of the buy-out, a printing operation, and rules nobody has written since.
Nothing on this page is a reason to buy a ticket.
Where these figures come from: Sharing Map · Return Ledger
Chernoff’s corner
Suppose the edge really were positive. Herman Chernoff found exactly that in the Massachusetts Numbers Game, wrote it up, and then explained at length why he was not going to bet on it. Here is his argument, run rather than described.
500 players. Each bets $1 a draw for 500 draws on a bet that genuinely pays 10% more than it should — a real edge, the kind this page has just spent its length demonstrating.
The expectation was on their side the whole time. Most of them still lost everything — the mean survives on a handful of large winners, and the median player is nowhere near it. An edge is not a plan; it is a statement about an average that a finite bankroll may never reach.
Nothing on this page changes your chance of winning. It only changes how many strangers you’d share with IF you win.
Every parameter, on the table
| Effect | Weight | Where it comes from |
|---|---|---|
| Numbers 1–31 — the birthday zone | 1.22× | Cook & Clotfelter (1993) on conscious selection; Farrell, Hartley, Lanot & Walker (2000) modelling UK number preferences. Dates cannot exceed 31, and birthdays are the commonest way people choose. |
| Numbers 1–12 — day and month overlap | 1.10× | Applied on top of the 1–31 elevation: numbers that can be either a day or a month are reachable by more date-based choices. |
| The number 7 | 1.15× | Consistently the most over-picked number in Western lotteries (Farrell et al. 2000; see also the cultural atlas). |
| Culturally lucky numbers 3, 8, 9 | 1.10× | Documented cultural preferences; 8 in particular in Chinese-speaking populations — the cultural atlas covers the traditions behind them. |
| The number 13, in Western markets | 0.75× | Under-picked in Anglo/Nordic markets (triskaidekaphobia). Note the reverse holds in Spain, so this parameter is scope-dependent. |
| Numbers above 31 | 0.86× | The complement of the birthday effect: numbers unreachable by any date are chosen markedly less (Cook & Clotfelter 1993). |
| Round numbers (multiples of 10) | 1.05× | Mild elevation for salient round values; the weakest effect in the table. |
| A long consecutive run | 2.20× | The 2008 Canadian draw 40-41-42-43-44-45 (bonus 43) produced 239 second-prize winners — evidence that “obvious” tickets are heavily played, not avoided. |
| Even spacing (an arithmetic pattern) | 1.60× | Baker & McHale (JRSS-A 2009) on combination-level preferences: players favour tickets that look designed. |
| Every number inside 1–31 | 1.40× | Cook & Clotfelter (1993): a ticket made entirely of date-reachable numbers sits in the most crowded region of the space. |
Illustrative model, calibrated to published studies — not live data.
The two layers underneath
A jackpot is pari-mutuel: the prize is fixed, and everyone holding the winning combination divides it. So if other people hold your numbers, you receive
of the prize. 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 combinations. Then the count holding yours is Poisson with mean
where is the number of tickets sold and is how many times likelier your particular combination is to be chosen than an indifferent picker would choose it. For a ticket nobody favours, .
Now the expectation. It looks as though it should need numerical work, and it does not:
The trick is in the middle step: 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 .
Two limits are worth holding onto. As the share tends to 1 — if nobody else is likely to hold your numbers, you keep the lot. And for large the share behaves like , 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.
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 chosen to reproduce those directions, and a ticket's is
where is the elementary symmetric polynomial of degree — the sum over all 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.
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$31$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 for every combination. The extra sharing eats most of the extra money: the jackpot contribution to expected value is approximately
and since grows with , 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.
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.
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.
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 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.
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.
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.