Facts

Facts about randomness, every one sourced

The numbers people get wrong at dinner, with the receipts attached: coins, dice, cards, lotteries, human habits and the machines that try to be unpredictable.

By Stuck at Home, LLC · October 5, 2026 · 44 facts, 61 sources

Numbers worth repeating at dinner — about coins, dice, cards, lotteries, people and the machines we trust to be unpredictable. Each card says where it came from.

Coins 4
Cards and dice 4
People 10
Probability surprises 4
Lotteries 7
Decided by chance 6
Machines 9

Coins

The simplest randomizer there is — and not quite as simple as it looks.

50.8%

A flipped coin tends to land the way it started

In 350,757 flips by 48 people, coins caught in the hand landed on the same side they started 50.8% of the time (95% interval 50.6–50.9%). Heads versus tails was dead even at 50.0%; the bias was all in the starting side, varied a lot between flippers, and shrank with practice. The study won the 2024 Ig Nobel Prize in Probability.

Flip a thousand →Sources: Fair coins tend to land on the same side they started — Bartoš et al., arXiv 2023 / JASA 2025 · Ig Nobel Prize winners — Improbable Research

≈51%

Physics predicted it first

In 2007 Persi Diaconis, Susan Holmes and Richard Montgomery used high-speed photography and the mechanics of a wobbling coin to predict that a vigorously flipped, hand-caught coin comes up as it started about 51% of the time. Sixteen years later the big flip study confirmed it.

Source: Dynamical Bias in the Coin Toss — SIAM Review 2007 (PDF)

1 in 6,000

An American nickel landing on its edge

Physicists Daniel Murray and Scott Teare dropped coin-like cylinders of different thicknesses, fitted a model and extrapolated: a tossed US nickel should land on its edge roughly once in 6,000 tries. An estimate from a model, not a count of actual nickels — but the only serious number anyone has.

Source: Probability of a tossed coin landing on edge — Physical Review E, 1993

<10%

Don’t spin a penny

Spinning a coin on a table is far less fair than flipping it. In data from a Berkeley class of 103 students, tossed pennies came up near 50% heads, but the same pennies spun on their edge leaned hard towards tails — several students’ coins came up heads less than 10% of the time. The bias depends on the edge shape and the exact centre of gravity.

Source: Dynamical Bias in the Coin Toss — SIAM Review 2007 (PDF)

Cards and dice

Big numbers hiding in small objects.

8 × 10⁶⁷

Ways to order a deck of cards

52! is roughly 80 million trillion trillion trillion trillion trillion — about 600 million billion times the estimated 10⁵⁰ atoms in the Earth, and unimaginably more than the 4.4 × 10¹⁷ seconds since the Big Bang. A properly shuffled deck is almost certainly in an order no deck has ever been in before.

Shuffle one →Sources: How many atoms are there in the world? — Jefferson Lab (archived) · Planck reveals an almost perfect Universe — ESA

Riffle shuffles to mix a deck

Dave Bayer and Persi Diaconis showed in 1992 that about (3/2)·log₂(n) riffles are needed to mix n cards — seven for a deck of 52. Fewer, and the original order is still detectable; more adds little. Measuring by information content instead, the Trefethens argued five do a fair job and six make mixing practically certain.

Watch the rainbow break up →Sources: Trailing the Dovetail Shuffle to its Lair — Annals of Applied Probability, 1992 · In Shuffling Cards, 7 Is Winning Number — New York Times, 1990 (archived)

≈10,000

Overhand shuffles to do the same job

The casual overhand shuffle — dribbling small packets from one hand to the other — is spectacularly inefficient. Diaconis and Jason Fulman put the number needed to mix a deck well at around ten thousand.

Source: Link between math and card shuffling — USC Dornsife

6 in 36

Why 7 wins with two dice

There are six ways to make 7 (1+6, 2+5, 3+4 and their mirror images) and only one way each to make 2 or 12. So 7 arrives one roll in six, the extremes one roll in thirty-six. Our own arithmetic; the full table is on the dice page.

Every total, worked out →Source: Dice — Wolfram MathWorld

People

We are terrible at being random, and we are bad at it in very consistent ways.

Blue · 7

The blue-seven phenomenon

Ask for a number from 0 to 9 and a colour, and the most common answers are 7 and blue. William Simon reported it from 490 college students in 1971 and, with Louis Primavera, from 533 schoolchildren in 1972, where the pair was “by far the most frequently written”. The papers give no percentages, so neither do we.

Sources: Number and Color Responses of Some College Students — Perceptual and Motor Skills, 1971 (record) · Investigation of the “Blue Seven Phenomenon” — Psychological Reports, 1972 (record)

37 · 73

The numbers that feel most random

In 2024 the YouTube channel Veritasium asked viewers for a random number from 1 to 100 and got about 200,000 answers. Setting aside the extremes and the joke picks 42 and 69, the standouts were 7, 73, 77 and 37; asked which number the fewest others would pick, people said 73 and 37, nearly tied. A self-selected online poll, not a study — but a very large one.

Source: Why is this number everywhere? — Veritasium, YouTube, June 2024 (figures as stated in the video)

≈80%

AI picks 7 even more than we do

Asked for a number from 1 to 10, several large language models — GPT-4o-mini, Phi-4 and Gemini 2.0 — answered 7 about 80% of the time across 75,600 calls in a 2025 study. Asked to guess a number from 1 to 50, ChatGPT, Claude Sonnet 4, Gemini 2.5 Flash and Llama 4 all said 27 in a June 2025 test. Models change with every version; the habit of having favourites has not.

Sources: Deterministic or probabilistic? The psychology of LLMs as random number generators — arXiv, 2025 · Ask a model to guess a number from 1 to 50 and it’s likely to answer 27 — The Register, June 2025

60% vs 50%

We switch too often

A truly random sequence of heads and tails changes side about half the time. The sequences people produce — and the ones they judge “most random” — switch about 60% of the time, a finding stable across decades of studies (Falk & Konold, 1997). Real randomness has more streaks than feels right.

Spot the real coin flips →Source: Making sense of randomness: implicit encoding as a basis for judgment — Falk & Konold, Psychological Review 1997 (PDF)

6 or 7

The longest streak in 100 coin flips

Most often 6 or 7 in a row: about 81% of 100-flip sequences contain a run of at least six, about 54% a run of at least seven. Mark Schilling’s rule of thumb is log₂(n) for the longest run of either side. The Hungarian educator Tamás Varga used it to catch students who faked their homework: invented sequences almost never contain the long runs real ones do.

Source: Long Run Predictions — Mark Schilling, Math Horizons 1994 (PDF)

1985 → 2018

The hot hand was real after all (probably)

A famous 1985 study of basketball shooting found no evidence that a hit makes the next shot likelier and called the “hot hand” an illusion. In 2018 Joshua Miller and Adam Sanjurjo proved the standard way of measuring streaks in short sequences is itself biased; correcting it, the original data point the other way. A randomness lesson about the people studying randomness.

Sources: The hot hand in basketball — Gilovich, Vallone & Tversky, 1985 (PDF) · Surprised by the Hot Hand Fallacy? — Miller & Sanjurjo, Econometrica 2018

1814 → 2016

The gambler’s fallacy, from Laplace to the courtroom

Laplace described it in 1814: crowds backing lottery numbers that hadn’t come up for a while, and expectant fathers growing anxious when the month had produced too many boys. In 2016 economists found the same reflex in US asylum judges, loan officers and baseball umpires — after a streak of one decision they leaned towards the opposite one, as if the sequence were owed a change.

Sources: A Philosophical Essay on Probabilities — Laplace, 1814 (Project Gutenberg) · Decision-Making under the Gambler’s Fallacy — Chen, Moskowitz & Shue, NBER 2016

537 bombs

The “clusters” that were just chance

Londoners in 1944 were sure the flying bombs fell in clusters. The actuary R. D. Clarke divided 144 km² of south London into 576 quarter-kilometre squares and counted: 229 squares hit never, 211 once, 93 twice, 35 three times, 7 four times, 1 five or more. The counts matched the Poisson distribution for purely random impacts almost exactly. Clusters are what randomness looks like.

Source: An Application of the Poisson Distribution — R. D. Clarke, Journal of the Institute of Actuaries 1946 (PDF)

10,000 letters

The puzzle that fooled the professors

When Marilyn vos Savant wrote in Parade in 1990 that a game-show contestant should switch doors — switching wins two times in three — she received about 10,000 letters, most disagreeing, close to 1,000 of them signed by PhDs. Paul Erdős reportedly refused to believe it until shown a computer simulation. Steve Selvin had posed the puzzle in 1975.

Sources: Behind Monty Hall’s Doors: Puzzle, Debate and Answer? — New York Times, 1991 (archived) · Game Show Problem — Marilyn vos Savant (archived)

Probability surprises

Numbers that are correct and still feel wrong.

23 people

A shared birthday is more likely than not

With 23 people in a room, the chance that at least two share a birthday is 50.7%; with 50 it is 97.0%, with 57 it is 99.0% and with 70, 99.9% (365 equally likely days, leap years ignored). First published by Richard von Mises in 1939; often credited to Harold Davenport, who posed it around 1927 but never published.

Why repeats come early →Source: Birthday Problem — Wolfram MathWorld

30.1%

How often a number starts with 1

In many real-world collections of numbers — river lengths, populations, invoices — the leading digit is 1 about 30.1% of the time and 9 only 4.6%. Simon Newcomb noticed in 1881 that the first pages of logarithm tables wore out fastest; Frank Benford tested it in 1938 on 20,229 numbers. Accountants now use it to spot invented figures.

Sources: Note on the Frequency of Use of the Different Digits in Natural Numbers — Newcomb, 1881 (record) · The Law of Anomalous Numbers — Benford, 1938 (record) · I’ve Got Your Number — Nigrini, Journal of Accountancy 1999

Hundreds a day

The law of truly large numbers

“With a large enough sample, any outrageous thing is likely to happen,” wrote Persi Diaconis and Frederick Mosteller in 1989. Their example: something that happens to one person in a million each day happens to hundreds of people every day in a country of 250 million. Coincidences are not rare; people are plentiful.

Source: Methods for Studying Coincidences — Diaconis & Mosteller, JASA 1989 (PDF)

1 a month

Your personal miracle rate

If a “miracle” is a one-in-a-million event, you should expect about one a month: we notice roughly one event per second for eight waking hours a day, which is about a million events a month. J. E. Littlewood supplied the one-in-a-million threshold in 1953; the once-a-month framing is Freeman Dyson’s, from a 2004 essay.

Source: One in a Million — Freeman Dyson, New York Review of Books 2004

Lotteries

Draws that repeated, numbers that everyone picked, and the arithmetic behind the ticket.

Twice in a row

The same six numbers, four days apart

Bulgaria’s national 6/42 lottery drew 4, 15, 23, 24, 35 and 42 on 6 September 2009 — and again, in a different order, on 10 September. Nobody won the first time; a record 18 people had the numbers the second time. A government-ordered investigation found no wrongdoing. Astonishing, and also exactly the kind of thing that eventually happens somewhere.

Sources: Bulgarian lottery repeat probed — BBC News, 2009 · Bulgaria’s identical lottery draw was coincidence — Reuters, 2009 (saved copy)

5 6 7 8 9 + 10

Twenty winners for a “suspicious” sequence

South Africa’s PowerBall draw of 1 December 2020 produced 5, 6, 7, 8, 9 and PowerBall 10. Twenty tickets shared the R114 million jackpot, about R5.7 million each. The public cried foul; the operator pointed out that runs like this are among the most commonly played lines in the world — which is why so many people had it.

Why patterns feel unlikely →Source: 20 people share “it’s not unusual” R114m PowerBall jackpot — TimesLIVE, 2020

7,059,052

The man who tried to buy every ticket

For the Virginia Lottery’s draw of 15 February 1992 — six numbers from 44, 7,059,052 combinations — a syndicate organised in part by Stefan Mandel set out to buy every one. It ran out of time about two million tickets short, held the only winning ticket anyway, and in March 1992 Virginia agreed to pay the $27 million jackpot.

Source: They lucked out in Va. lottery — AP via Tampa Bay Times, 1992

≈2,100 entries

People who play 1-2-3-4-5-6

In October 2010 New Zealand Lotteries checked the previous week’s Lotto draw and found around 2,100 entries using 1, 2, 3, 4, 5, 6. The line is exactly as likely as any other — about one in 3.8 million, the operator said — but a win would have been split about 2,100 ways. The often-quoted British figure of 10,000 could not be traced to any operator statement, so it isn’t here.

Source: Some numbers not so lucky for some — New Zealand Lotteries Commission release via Scoop, 8 October 2010

5.5 × 10²⁶

Possible American bingo cards

With 75 balls — B 1–15, I 16–30, N 31–45, G 46–60, O 61–75 — and five numbers per column (four in N, around the free square), there are 552,446,474,061,128,648,601,600,000 distinct cards. British bingo uses 90 balls instead. Our own count from the official layout.

Source: Minnesota Rules 2001, chapter 7861 — bingo equipment

0.750–0.775 in

How perfect a casino die must be

US regulators require craps dice to be perfect transparent cubes between 0.750 and 0.775 inch on a side, with square edges, flat faces and flush spots — each spot drilled out and refilled with material of exactly the same weight, so no face is heavier. Ohio requires every spot to be drilled to the same depth within 0.0004 inch.

Roll one anyway →Sources: Mich. Admin. Code R 432.1814 — Dice specifications (Cornell LII) · Ohio Administrative Code 3772-11-23 — dice

Decided by chance

Cities, flights and parliaments that came down to a toss or a draw.

2 of 3

Portland, Oregon was named by a coin toss

In 1845 Asa Lovejoy of Boston and Francis Pettygrove of Portland, Maine, settled the name of their new town over dinner with a best-of-three toss. Pettygrove won. The coin, an 1835 copper cent now called the Portland Penny, is on display at the Oregon Historical Society. Lose the toss and the city is called Boston.

Source: Portland Penny — The Oregon Encyclopedia (archived)

14 Dec 1903

Wilbur won the toss and lost the flight

The Wright brothers flipped a coin for the first attempt at powered flight. Wilbur won and tried on 14 December 1903; the attempt failed. When the Flyer was ready again on 17 December it was Orville’s turn, and his is the flight in the history books.

Source: Frequently Asked Questions — Wright Brothers National Memorial, National Park Service

51–49

A legislature decided by drawing a name from a bowl

Virginia’s 2017 race for the 94th House of Delegates district ended in an exact tie after a recount. On 4 January 2018 each candidate’s name was sealed in a film canister and placed in a blue stoneware bowl; the elections board chairman drew David Yancey’s. That one draw kept Republicans at 51–49 and gave them control of the chamber.

Source: Republican wins tied Virginia House race after random drawing — NBC News, 2018

99 citizens

A constitution changed by people chosen at random

Ireland’s Citizens’ Assembly of 2016–18 brought 99 randomly selected members of the public together under a Supreme Court judge to deliberate on, among other things, the constitutional near-ban on abortion. Parliament took up its report, and the referendum of 25 May 2018 removed the Eighth Amendment. Sortition — the Athenian idea — at work in the twenty-first century.

How Athens did it →Source: 2016–2018 Citizens’ Assembly — citizensassembly.ie

5 pages of “s”

The infinite-monkey experiment, actually run

In 2003 staff and students from the University of Plymouth put a computer in the enclosure of six Sulawesi crested macaques at Paignton Zoo. After a month the monkeys had produced five pages of text, mostly the letter s, and had partially destroyed the keyboard and used it as a lavatory. Shakespeare remained unwritten.

Source: No words to describe monkeys’ play — BBC News, 2003

Unproven

Nobody has proved that π’s digits are random

Across trillions of computed digits, every digit and digit string of π turns up about as often as chance predicts. But no one has proved π is “normal” — that this holds for ever — and, in fact, no naturally occurring mathematical constant has been proved normal in even one number base.

Sources: Normal Number — Wolfram MathWorld · Are the Digits of Pi Random? — Berkeley Lab

Machines

What it takes to make a computer, or a pile of ping-pong balls, unpredictable.

16 minutes

ERNIE’s first winning number

Britain’s first Premium Bond draw, on 1 June 1957, used ERNIE — a machine built by Colossus veterans that drew randomness from electrons moving through neon gas. It took 16 minutes to produce the first winning number and more than 55 hours to complete the draw. ERNIE 5, which took over in 2019, uses quantum light and finished a month’s draw in 12 minutes.

Sources: GPO ERNIE I — Science Museum Group · Premium Bonds take a quantum leap — NS&I, 2019 (archived)

15 planes

Where IBM’s “random” numbers actually lived

RANDU, the generator IBM shipped with its System/360 computers in the 1960s, had a flaw you can see: plot any three consecutive outputs as a point in space and every point lands on one of just 15 parallel planes. George Marsaglia’s 1968 paper exposed it, and the same lattice lurks in every generator of its kind.

Source: Random Numbers Fall Mainly in the Planes — Marsaglia, PNAS 1968

111 steps

The longest a four-digit middle-square run can last

Von Neumann’s square-and-keep-the-middle recipe dies fast. We ran all 10,000 four-digit seeds: every one falls into a loop, 2,263 of them freezing on a single number and the rest cycling through four. The longest run before a repeat — from seed 6239 — is 111 steps. Our own measurement; you can reproduce it on the story page.

Run a seed →Source: The experiment, in the page’s own code (lab-model.mjs)

25 seconds

To break Netscape’s secure connections in 1995

Netscape Navigator seeded its encryption keys from the time of day and two process IDs. Two Berkeley students showed that with an account on the same machine, trying every candidate seed took about 25 seconds; from outside, minutes. The cipher was fine. The randomness feeding it was not.

Source: Randomness and the Netscape Browser — Goldberg & Wagner, 1996

32,767

Possible keys per architecture, Debian 2006–2008

Two lines commented out to silence a debugging tool removed the entropy from Debian’s OpenSSL. For twenty months every key was seeded by nothing but the process ID — 32,767 possibilities — so SSH, VPN and web keys could simply be listed. Found in 2008; advisory DSA-1571.

Source: DSA-1571-1 — Debian Security Advisory (archived)

2¹⁹⁹³⁷ − 1

The period of the Mersenne Twister

The generator behind Python’s, R’s, Ruby’s and MATLAB’s default random numbers repeats only after about 4.3 × 10⁶⁰⁰¹ values. Long is not the same as secret: its whole state is 624 words, so a long enough run of outputs predicts the rest.

Sources: Mersenne Twister — Matsumoto & Nishimura, 1998 (PDF) · random — Python 3 documentation

≈100 lamps

Cloudflare’s lobby, San Francisco

A camera films a wall of about a hundred lava lamps and feeds the video into a secure generator as an extra source of randomness for Cloudflare’s servers. Other offices contribute double pendulums (London), a small uranium pellet (Singapore), rainbow mobiles (Austin) and fifty wave machines (Lisbon).

Photograph our lamp wall →Sources: Randomness 101: LavaRand in Production — Cloudflare, 2017 · Harnessing chaos in Cloudflare offices — Cloudflare

512 bits / 60 s

Public randomness, on a schedule

NIST’s Randomness Beacon publishes 512 fresh random bits every minute, signed and chained to the pulse before so the past can’t be rewritten; the League of Entropy’s drand network emits a jointly signed round every 30 seconds, and its quicknet chain every 3. Both are for draws that must be checkable by strangers — and both warn you never to use the values as secret keys.

Sources: Interoperable Randomness Beacons — NIST · About drand — drand.love

71,313 bits

Certified by a quantum computer

In March 2025 a 56-qubit trapped-ion machine, treated as an untrusted source, produced numbers that classical supercomputers running at 1.1 exaFLOPS then certified as 71,313 bits of genuine entropy. Seven years earlier NIST had certified 1,024 bits with entangled photons and a loophole-free Bell test — randomness guaranteed by physics rather than by trust.

Sources: Harnessing Quantum Computing for Certified Randomness — JPMorganChase, 2025 · Experimentally Generated Randomness Certified by the Impossibility of Superluminal Signals — arXiv, 2018

Make one more random fact.

The next number nobody could have predicted is one tap away.

Pick a random number

Where we looked things up