The memoryless machine

“That number is due.”

It is the most persistent feeling in the game, and Dakota Cash has a number to hang it on right now. Here is the board. Then here is the era’s own history, asked whether waiting has ever made the slightest difference.

Somebody has to be the straggler

Longest current waits across 35 numbers and 1,725 draws.

A given number waiting 25 draws happens 2.1% of the time. But the chance that some number among all 35 is at least this overdue right now is 52.8%. The board is never empty. Somebody always has to be at the top of it.

Number 24, draw by draw

Drawn 242 times in 1,725 draws · average wait 7.1 draws · longest completed wait 40 · waiting 14 draws now

a completed wait — darker is longer the wait still running — a lower bound, not a finished gap

Does waiting help?

Dakota Cash5/35 era · since Apr 20101,725 draws

Scroll the chart sideways →

If “due” were real, this line would climb to the right. Each point is every occasion in 1,725 draws when a number had waited at least that long, and how often it turned up in the very next draw. It wanders, because every point is a sample — but 1 of 41 points sit outside the rule’s 95% interval, about the 2 chance alone predicts. The whiskers are those 95% intervals; the gold point gathers every longer wait there is, and the chart stops where too few occasions remain to say anything honest.

How long numbers actually wait

Dakota Cash5/35 era · since Apr 20101,725 draws

Scroll the chart sideways →

Most waits are short. The long ones are what people notice.

Every completed wait in this era, against the geometric curve chance predicts. Switch to the log scale to watch the tail — the part nobody checks — fall on the same line as the rest.

The record book

Against the longest drought a provably fair machine produces over the same number of draws (200 simulated eras).

The band is the range for a fair machine’s single worst drought, so only the top row is measured against it. The rows beneath are this era’s runners-up, and they fall short of it by construction.

17 waited 64 draws — the era’s worst. A fair machine’s worst drought over this many draws lands between 53–79, 62. It is comfortably inside the band. Nothing to explain.

Now the same thing, with nothing behind it

The right-hand era never happened. It came out of a random number generator moments ago, so there is provably no bias, no worn ball, no pattern of any kind in it. Compare the two boards.

The real era

Somebody has to be the straggler

Longest current waits across 35 numbers and 1,725 draws.

1325drawslast seen 2026-07-18 · 88 days3.6× the average2.1%how likely
219drawslast seen 2026-08-08 · 67 days2.7× the average5.3%how likely
3216drawslast seen 2026-08-19 · 56 days2.3× the average8.5%how likely
715drawslast seen 2026-08-22 · 53 days2.1× the average9.9%how likely
2414drawslast seen 2026-08-26 · 49 days2.0× the average11.6%how likely
2813drawslast seen 2026-08-29 · 46 days1.9× the average13.5%how likely

Real records

Against the longest drought a provably fair machine produces over the same number of draws (200 simulated eras).

The band is the range for a fair machine’s single worst drought, so only the top row is measured against it. The rows beneath are this era’s runners-up, and they fall short of it by construction.

17 waited 64 draws — the era’s worst. A fair machine’s worst drought over this many draws lands between 53–79, 62. It is comfortably inside the band. Nothing to explain.

A machine we know is fair

Somebody has to be the straggler

Longest current waits across 35 numbers and 1,725 draws.

2521drawslast seen 2026-08-01 · 74 days3.0× the average3.9%how likely
2716drawslast seen 2026-08-19 · 56 days2.3× the average8.5%how likely
614drawslast seen 2026-08-26 · 49 days2.0× the average11.6%how likely
3411drawslast seen 2026-09-05 · 39 days1.6× the average18.3%how likely
1110drawslast seen 2026-09-09 · 35 days1.4× the average21.4%how likely
39drawslast seen 2026-09-12 · 32 days1.3× the average25.0%how likely

A given number waiting 21 draws happens 3.9% of the time. But the chance that some number among all 35 is at least this overdue right now is 75.4%. The board is never empty. Somebody always has to be at the top of it.

Simulated records

Against the longest drought a provably fair machine produces over the same number of draws (200 simulated eras).

The band is the range for a fair machine’s single worst drought, so only the top row is measured against it. The rows beneath are this era’s runners-up, and they fall short of it by construction.

11 waited 68 draws — the era’s worst. A fair machine’s worst drought over this many draws lands between 53–79, 62. It is comfortably inside the band. Nothing to explain.

Both boards have a straggler. Both have a drought that would make a headline. Drama is not evidence — it is what watching 35 numbers for 1,725 draws looks like.

The two layers underneath

The maths, in plain language

Every draw is a fresh coin

Fix on one number — say 24. On any given draw it either comes out or it does not, and because 5 of the 35 numbers are drawn, the chance it comes out is

p = \frac{k}{N}

For Dakota Cash that is 5 in 35 — 14.29% — and it is the same on every draw, forever.

The wait between two appearances is then a run of failures followed by a success — the geometric distribution:

P(\text{gap} = g) = p(1-p)^{g-1}, \qquad P(\text{gap} \ge g) = (1-p)^{g-1}

The mean is not the typical wait

E[\text{gap}] = \frac{N}{k}

which here is 7.00 draws — but the median wait is only 5 draws. Half of all waits are shorter than that. The average is dragged upward by a long tail, so "this number is overdue, it usually comes every 7.0 draws" misreads its own statistic: most waits are shorter than the mean, and the ones that are longer can be very much longer.

Memorylessness — the whole argument

Suppose a number has already failed to appear for $a$ draws. What is the chance it waits at least $b$ more?

P(X > a+b \mid X > a) = \frac{(1-p)^{a+b}}{(1-p)^{a}} = (1-p)^{b} = P(X > b)

The $a$ cancels. Completely. The waiting simply does not appear in the answer, which means

P(\text{appears next} \mid \text{gap} = 50) = P(\text{appears next} \mid \text{gap} = 0) = \frac{k}{N}

The geometric is the only discrete distribution with this property, and the lottery is one of the few places in ordinary life where it genuinely holds. The Flatline chart above is this identity wearing real data: at every wait length, from one draw to 41, the observed return rate sits on 14.29%.

Why a long drought is guaranteed

Watch $N$ numbers for $D$ draws and you are not running one waiting-time experiment, you are running $N \cdot D \cdot p$ of them. The longest wait you should expect grows like

\max \text{gap} \approx \frac{\ln(N \cdot D \cdot p)}{-\ln(1-p)}

For this era that is about 59 draws, against an average wait of 7.0. A drought several times the average is not a symptom. It is arithmetic — the same look-elsewhere effect that makes a striking coincidence certain somewhere as soon as you look in enough places.

The Record Book's band puts a number on that directly, by simulating 200 fair eras of exactly this length and asking how long their worst drought ran.

Two honest caveats

The board's probability is approximate. "The chance that some number is at least this overdue" is computed as

1 - \prod_{i=1}^{N} P(\text{gap}_i < g) = 1 - \left(1 - (1-p)^{g-1}\right)^{N}

which treats the 35 numbers as independent. They are not: exactly 5 appear in every draw, so if one number is absent another must be present, and the waits are mildly negatively correlated. The approximation therefore overstates the figure slightly. It is close enough to make its point — which is that the figure is near-certain — and the point survives the correction comfortably.

The current wait is censored. A number that has been absent for 25 draws has not had a gap of 25; it has had a gap of at least 25, and nobody yet knows what the finished number will be. That is why the board reads $\ge$ , why open waits are drawn differently in the timelines, and why they are never averaged in with completed ones — doing so would understate every average on this page while overstating the record.

The story behind it

Monte Carlo, 18 August 1913

At a roulette table in the Monte Carlo Casino, black came up. Then black again. By the fifteenth black the room had noticed, and money began moving onto red — because red was surely due now. By the twentieth the crowd was several deep. Gamblers doubled and redoubled, certain that each spin made the correction more inevitable.

Black came up twenty-six times in a row. Millions of francs were lost, and almost all of them were lost in the last ten spins, by people who were not betting on red so much as betting against a run they believed could not continue.

The wheel had no way to know. Each spin was its own event, with the same slightly-worse-than-even odds as the first, and the chance of that particular run was around one in 66 million — a number that sounds impossible until you remember how many thousands of tables have spun how many millions of times since. Somewhere, some evening, twenty-six blacks was always going to happen. The gambler's fallacy is still called the Monte Carlo fallacy because of that night.

Italy, 2003 to 2005

Italy's Lotto draws numbers on ten regional wheels. On the Venice wheel, the number 53 did not appear for close to two years — 182 consecutive draws.

The country noticed. Newspapers ran counters. Betting on 53 became a national habit and then a national compulsion; an estimated €3.5 billion was staked on the number as the drought lengthened, much of it by people who could not afford to lose it. Consumer groups began describing it publicly as an epidemic.

Before 53 was drawn on 9 February 2005, the deaths of at least four people were linked in the Italian press to the frenzy: a woman in Tuscany who drowned herself after losing the family's savings; a man in Signa who shot his wife and son and then himself; others whose debts had followed the same number down. The national consumer association Codacons called for the number to be withdrawn from the game.

The wheel had not been holding 53 back. Over 182 draws with five numbers from ninety, a specific number failing to appear has a probability of roughly one in five hundred — unusual for that number, on that wheel, but there were ten wheels and ninety numbers, and a drought of that length somewhere among them was close to expected. What made those two years catastrophic was not the mathematics. It was the belief that a machine which had withheld something for long enough owed it back.

This is the point at which the gambler's fallacy stops being an interesting quirk of human reasoning.

Why the intuition exists at all

The instinct is not stupid. It is trained, and it is trained on a world where it works.

If a bus is scheduled every ten minutes and you have been waiting twelve, you genuinely are closer to a bus than when you arrived — there is a timetable, a depot, a driver. Waiting is informative almost everywhere: for the kettle, the lift, the post, the end of a song. The longer something has been coming, the sooner it arrives.

The lottery is one of the very few systems in ordinary life with a truly memoryless timetable. Our intuitions were built in the other world, and they travel here without any warning that they have stopped applying.

The one real effect

History cannot tip the machine. But the belief in "due" numbers is not harmless — because it changes what people play, and what people play changes what a win is worth.

When thousands of players pile onto the same overdue number, they are not making that number more likely. They are making it more crowded: if it does come up, the prize is split more ways. The mirror image is just as well documented — Clotfelter and Cook found that bets on a number fall sharply right after it wins, since it has just "had its turn".

So the fallacy does have an exploitable edge, and it belongs to whoever does not hold it. Not by predicting the draw, which nobody can do, but by not standing where the crowd is standing.

Monte Carlo Casino, 18 August 1913: 26 consecutive blacks — the origin of the “Monte Carlo fallacy”.
Italy, Venice wheel: number 53 absent for 182 draws to 9 February 2005; ≈€3.5bn staked; bankruptcies and deaths reported in the Italian and international press (incl. The Guardian, February 2005).
Clotfelter, C. & Cook, P. “The ‘Gambler’s Fallacy’ in Lottery Play.” Management Science 39(12), 1993.

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Reading the draws and working out what chance predicts.

The memoryless machine

“That number is due.”

Somebody has to be the straggler

Longest current waits across 35 numbers and 1,725 draws.

Number 24, draw by draw

Drawn 242 times in 1,725 draws · average wait 7.1 draws · longest completed wait 40 · waiting 14 draws now

a completed wait — darker is longer the wait still running — a lower bound, not a finished gap

Does waiting help?

Dakota Cash5/35 era · since Apr 20101,725 draws

Scroll the chart sideways →

How long numbers actually wait

Dakota Cash5/35 era · since Apr 20101,725 draws

Scroll the chart sideways →

Most waits are short. The long ones are what people notice.

Every completed wait in this era, against the geometric curve chance predicts. Switch to the log scale to watch the tail — the part nobody checks — fall on the same line as the rest.

The record book

Against the longest drought a provably fair machine produces over the same number of draws (200 simulated eras).

The band is the range for a fair machine’s single worst drought, so only the top row is measured against it. The rows beneath are this era’s runners-up, and they fall short of it by construction.

17 waited 64 draws — the era’s worst. A fair machine’s worst drought over this many draws lands between 53–79, 62. It is comfortably inside the band. Nothing to explain.

Now the same thing, with nothing behind it

The right-hand era never happened. It came out of a random number generator moments ago, so there is provably no bias, no worn ball, no pattern of any kind in it. Compare the two boards.

The real era

Somebody has to be the straggler

Longest current waits across 35 numbers and 1,725 draws.

1325drawslast seen 2026-07-18 · 88 days3.6× the average2.1%how likely
219drawslast seen 2026-08-08 · 67 days2.7× the average5.3%how likely
3216drawslast seen 2026-08-19 · 56 days2.3× the average8.5%how likely
715drawslast seen 2026-08-22 · 53 days2.1× the average9.9%how likely
2414drawslast seen 2026-08-26 · 49 days2.0× the average11.6%how likely
2813drawslast seen 2026-08-29 · 46 days1.9× the average13.5%how likely

Real records

Against the longest drought a provably fair machine produces over the same number of draws (200 simulated eras).

The band is the range for a fair machine’s single worst drought, so only the top row is measured against it. The rows beneath are this era’s runners-up, and they fall short of it by construction.

17 waited 64 draws — the era’s worst. A fair machine’s worst drought over this many draws lands between 53–79, 62. It is comfortably inside the band. Nothing to explain.

A machine we know is fair

Somebody has to be the straggler

Longest current waits across 35 numbers and 1,725 draws.

2521drawslast seen 2026-08-01 · 74 days3.0× the average3.9%how likely
2716drawslast seen 2026-08-19 · 56 days2.3× the average8.5%how likely
614drawslast seen 2026-08-26 · 49 days2.0× the average11.6%how likely
3411drawslast seen 2026-09-05 · 39 days1.6× the average18.3%how likely
1110drawslast seen 2026-09-09 · 35 days1.4× the average21.4%how likely
39drawslast seen 2026-09-12 · 32 days1.3× the average25.0%how likely

Simulated records

Against the longest drought a provably fair machine produces over the same number of draws (200 simulated eras).

The band is the range for a fair machine’s single worst drought, so only the top row is measured against it. The rows beneath are this era’s runners-up, and they fall short of it by construction.

11 waited 68 draws — the era’s worst. A fair machine’s worst drought over this many draws lands between 53–79, 62. It is comfortably inside the band. Nothing to explain.

Both boards have a straggler. Both have a drought that would make a headline. Drama is not evidence — it is what watching 35 numbers for 1,725 draws looks like.

The two layers underneath

The maths, in plain language

Every draw is a fresh coin

Fix on one number — say 24. On any given draw it either comes out or it does not, and because 5 of the 35 numbers are drawn, the chance it comes out is

p = \frac{k}{N}

For Dakota Cash that is 5 in 35 — 14.29% — and it is the same on every draw, forever.

The wait between two appearances is then a run of failures followed by a success — the geometric distribution:

P(\text{gap} = g) = p(1-p)^{g-1}, \qquad P(\text{gap} \ge g) = (1-p)^{g-1}

The mean is not the typical wait

E[\text{gap}] = \frac{N}{k}

Memorylessness — the whole argument

Suppose a number has already failed to appear for $a$ draws. What is the chance it waits at least $b$ more?

P(X > a+b \mid X > a) = \frac{(1-p)^{a+b}}{(1-p)^{a}} = (1-p)^{b} = P(X > b)

The $a$ cancels. Completely. The waiting simply does not appear in the answer, which means

P(\text{appears next} \mid \text{gap} = 50) = P(\text{appears next} \mid \text{gap} = 0) = \frac{k}{N}

Why a long drought is guaranteed

Watch $N$ numbers for $D$ draws and you are not running one waiting-time experiment, you are running $N \cdot D \cdot p$ of them. The longest wait you should expect grows like

\max \text{gap} \approx \frac{\ln(N \cdot D \cdot p)}{-\ln(1-p)}

The Record Book's band puts a number on that directly, by simulating 200 fair eras of exactly this length and asking how long their worst drought ran.

Two honest caveats

The board's probability is approximate. "The chance that some number is at least this overdue" is computed as

1 - \prod_{i=1}^{N} P(\text{gap}_i < g) = 1 - \left(1 - (1-p)^{g-1}\right)^{N}

The story behind it

Monte Carlo, 18 August 1913

Italy, 2003 to 2005

Italy's Lotto draws numbers on ten regional wheels. On the Venice wheel, the number 53 did not appear for close to two years — 182 consecutive draws.

This is the point at which the gambler's fallacy stops being an interesting quirk of human reasoning.

Why the intuition exists at all

The instinct is not stupid. It is trained, and it is trained on a world where it works.

The one real effect

History cannot tip the machine. But the belief in "due" numbers is not harmless — because it changes what people play, and what people play changes what a win is worth.

So the fallacy does have an exploitable edge, and it belongs to whoever does not hold it. Not by predicting the draw, which nobody can do, but by not standing where the crowd is standing.

Monte Carlo Casino, 18 August 1913: 26 consecutive blacks — the origin of the “Monte Carlo fallacy”.
Italy, Venice wheel: number 53 absent for 182 draws to 9 February 2005; ≈€3.5bn staked; bankruptcies and deaths reported in the Italian and international press (incl. The Guardian, February 2005).
Clotfelter, C. & Cook, P. “The ‘Gambler’s Fallacy’ in Lottery Play.” Management Science 39(12), 1993.