50 Fun Facts About Probability

How many people in a room are needed before a shared birthday between two of them is more likely than not?

Just 23 people are enough, because every pair in the room is a separate chance at a match and a group that size holds hundreds of pairs. Most people guess well over a hundred.

In the Monty Hall problem, what chance of winning the car does switching give once the host reveals a goat?

Switching wins 2/3 of the time, because it loses only when the first pick was already the car, a one-in-three event. Pigeons trained on the puzzle learn to switch faster than people do.

What do statisticians call the belief that a fair coin, after ten heads in a row, is 'due' to land tails?

It is the gambler's fallacy, also called the Monte Carlo fallacy, and it runs backwards: past flips leave no trace on a fair coin, so tails stays at exactly one half. The 'hot hand' is the mirror-image error.

How many riffle shuffles did Bayer and Diaconis show a 52-card deck needs before its order is effectively random?

Seven riffle shuffles is the finding of the 1992 Bayer–Diaconis paper, which showed a deck is nowhere near random after five and essentially perfect after 7. A casino that shuffles less leaves patterns a sharp player can exploit.

Roughly how many different orderings can a shuffled 52-card deck take, the number mathematicians write as 52 factorial?

A 52-card deck has 52! ≈ 8 × 10^67 orderings, so any properly shuffled deck is almost certainly in an order no deck in history has ever been in before.

What are the jackpot odds for a Powerball ticket matching five white balls from 69 and the red ball from 26?

The odds are 1 in 292,201,338, and they were lengthened deliberately in 2015 so jackpots would roll over into headline-grabbing billions more often. The chance of winning any prize at all is about 1 in 25.

Per Persi Diaconis's physics model, with what probability does a caught coin land the same way up it started?

The same-side bias is about 51%, because a tossed coin wobbles as it spins and spends slightly longer with its starting face up. Diaconis measured it with a strobe light and a coin painted black on one side.

Which physicist told Max Born in a 1926 letter that he was convinced God 'is not playing at dice'?

Albert Einstein was rejecting Born's proposal that quantum mechanics predicts only probabilities with no underlying cause. He never accepted it, and his argument with Bohr ran on for decades.

How many ways can a 63-game March Madness bracket be filled in if every game is treated as a coin flip?

There are 2^63, about 9.2 × 10^18 or 9.2 quintillion, possible brackets, more than the estimated 7.5 quintillion grains of sand on Earth. Knowing the sport only cuts the NCAA's estimate to around 1 in 120 billion.

In the secretary problem, what share of candidates do you reject before taking the first one better than all of them?

Rejecting the first 37% — one divided by e — and then taking the next best-so-far candidate lands the single best applicant about 37% of the time, whether there are 100 or 100 million of them.

With two fair dice, what is the probability of a double six, the only way to throw a total of twelve?

A double six comes up 1/36 of the time, since each die has six faces and the pair makes 36 equally likely combinations. A total of seven, by contrast, can be made six different ways.

What is the probability that three fair coins all land heads, one of the two outcomes that void a three-way flip?

All heads happens 1/8 of the time (½ × ½ × ½), which is why the three-way coin flip used to pick an odd one out works on 75% of attempts and is simply repeated otherwise.

What is the expected value of a single roll of a fair six-sided die, the average seen over thousands of throws?

The expected value is 3.5, a result the die can never actually show, because each face from one to six is equally likely and their mean is 21 divided by 6. A pair of dice averages exactly double that.

If 1 driver in 1,000 is drunk and a breathalyser flags 5% of sober ones, how likely is a flagged driver drunk?

Only about 2%: among 1,000 drivers the device flags roughly 50 sober people for every one who is truly drunk, a base-rate effect that also makes rare-disease screening positives mostly false alarms.

Which solicitor's 1999 murder conviction over her two sons' deaths hinged on a discredited '1 in 73 million' statistic?

Sally Clark's jury heard paediatrician Roy Meadow claim two cot deaths in one family had odds of 1 in 73 million, a figure that wrongly treated the deaths as independent. Her convictions were quashed in 2003.

Which sport did Gilovich, Vallone and Tversky study in the 1985 paper that called the 'hot hand' an illusion?

Basketball shooting was the test case: the paper found no evidence that a player who has just sunk several shots is likelier to sink the next. A 2018 reanalysis argued the original data did show streaks after all.

What house edge does a European single-zero roulette wheel carry, about half the American double-zero figure?

The edge is 2.7%, one pocket in 37, and it applies to every bet on the layout, which is why no staking plan can beat the wheel in the long run. The American wheel's extra 00 pushes it to 5.26%.

At Monte Carlo in 1913, how many spins in a row came up black while bettors lost millions of francs backing red?

Black came up 26 times running, a streak with odds of roughly 1 in 68 million, and the crowd kept piling onto red in the belief that it had to end — the textbook case of the error named after that casino.

Which game show, where players traded prizes for what hid behind numbered doors, gave the Monty Hall problem its name?

Let's Make a Deal was hosted by Monty Hall from its 1963 debut, and its audience 'traders' gambling on unseen prizes inspired the three-door puzzle. Hall himself noted that real play on the show never followed the puzzle's rules.

In a normal bell curve, what share of values fall within one standard deviation either side of the mean?

About 68% lies within one standard deviation, 95% within two and 99.7% within three — the empirical rule that lets a single number say how unusual a height, a test score or a blood reading is.

Which Toulouse lawyer's 1654 letters with Blaise Pascal on a gambling problem founded probability theory?

Pierre de Fermat, who treated mathematics as a hobby beside his legal career, worked out with Pascal how to split the stakes of an interrupted game, the 'problem of points'. His famous Last Theorem was a margin note.

Which man of letters brought the 'problem of points', on splitting an unfinished game's pot, to Pascal in 1654?

The Chevalier de Méré, born Antoine Gombaud, was a writer and salon wit whose questions about dice and stakes drew Pascal into the subject. Within months Pascal and a correspondent in Toulouse had its foundations.

Which Italian polymath's Liber de ludo aleae, written around 1564, was the first systematic treatment of probability?

Gerolamo Cardano kept himself solvent at the gaming table, and his book, unpublished until 1663, includes a section on effective cheating beside the first definition of odds as favourable against unfavourable outcomes.

Which Dutch scientist, famed for the pendulum clock, wrote the first published treatise on probability, in 1657?

Christiaan Huygens took up games of chance after a 1655 visit to Paris, where he heard of the Pascal correspondence, and his De Ratiociniis in Ludo Aleae set out the idea of expected value, the average payoff of a bet.

Which Swiss mathematician's posthumous Ars Conjectandi of 1713 proved the law of large numbers, his 'golden theorem'?

Jacob Bernoulli spent over 20 years proving that a coin's observed share of heads settles toward its true probability as tosses mount, and the book appeared eight years after his death. His nephew Daniel gave fluids their principle.

Which Huguenot exile in London described the bell curve in 1733 and wrote The Doctrine of Chances?

Abraham de Moivre's 1733 result was the first statement of the normal curve as an approximation to the binomial, and legend says he predicted the date of his own death by noticing he slept 15 minutes longer each night.

Thomas Bayes, whose theorem updates a probability on new evidence, earned his living in what profession?

Bayes was a Presbyterian minister at the Mount Sion Chapel in Tunbridge Wells, and his theorem was never published in his lifetime; it surfaced only after his death in 1761.

Which Welsh philosopher edited and published Bayes's essay in 1763, two years after its author's death?

Richard Price, who also helped found actuarial science and championed American independence, sent the paper to the Royal Society with his own introduction; the Bayes–Price theorem would be a fairer name.

Which Soviet mathematician's 1933 Foundations of the Theory of Probability gave the subject its modern axioms?

Kolmogorov built probability on measure theory — an event is a set and its probability a measure that sums to one — and the same man went on to found the mathematics of turbulence and of algorithmic complexity.

Which Los Alamos mathematician dreamed up the Monte Carlo method in 1946 while playing solitaire during an illness?

Stanisław Ulam wondered what the odds were that a Canfield solitaire would come out, and decided to just lay out a hundred hands and count; the code name came from the casino where his uncle gambled with borrowed money.

Which mathematician, interned in Nazi-occupied Denmark, tossed a coin 10,000 times to test the law of large numbers?

John Kerrich, raised in South Africa and caught visiting in-laws in Copenhagen in April 1940, recorded 5,067 heads — a proportion that crept ever closer to one half as the tosses mounted, exactly as the theorem predicts.

Which Victorian polymath invented the 'bean machine', a pegboard down which balls fall to pile up in a bell curve?

Francis Galton, Darwin's half-cousin, presented the board in 1874 to show the central limit theorem in action; he also coined 'regression to the mean', devised fingerprint classification and, less happily, eugenics.

Which statistician's 'lady tasting tea' experiment tested Muriel Bristol's claim to tell if milk was poured first?

Ronald Fisher gave her eight cups, four of each kind, in random order; she got all eight right, something chance alone would manage only once in 70 tries — the experiment that introduced the null hypothesis.

Who built a life table from Breslau death records in 1693, letting the English government price life annuities by age?

Edmond Halley used records sent from Breslau by Caspar Neumann to work out how many people of each age survive, letting the government sell life annuities at a rate matched to the buyer's age — the root of actuarial science.

Which Danish engineer, sizing the Copenhagen telephone company's circuits, founded queueing theory in 1909?

Agner Erlang's name is now the unit of telephone traffic, and the Erlang formulas still size call centres and computer networks; his 1909 paper asked how many circuits keep callers from hearing the busy tone.

Which distribution did Ladislaus Bortkiewicz famously fit to Prussian cavalrymen killed by horse kicks in 1898?

The Poisson distribution describes rare events in a fixed stretch of time or space, and the horse-kick deaths fitted it so well the example still opens textbooks; it also models radioactive decays and goals in a football match.

Which MIT mathematician's 1962 book Beat the Dealer proved blackjack could be beaten by counting cards?

Edward Thorp ran the strategy on an IBM 704 before testing it in Reno and Las Vegas in disguise, and with Claude Shannon built the first wearable computer to predict roulette — a device Nevada outlawed in 1985.

In what ratio did yellow and green peas reappear in the second generation of Gregor Mendel's crosses?

The 3:1 ratio arises because each parent passes on one of two gene versions at random, so a quarter of offspring get two copies of the recessive green. Mendel's counts were later judged suspiciously close to perfect.

How many other people must be in a room before one of them more likely than not shares your own birthday?

It takes 253 other people, far more than for any two sharing, because every match must now involve you: each person is compared with you once, instead of every pair in the room being compared.

Which Parade columnist, once Guinness's highest-IQ holder, drew some 10,000 letters with her 1990 Monty Hall answer?

Marilyn vos Savant told readers to switch, and around 10,000 wrote in to say she was wrong; she was right, and about half of the published letter-writers later admitted they had changed their minds.

Which university's 1973 admissions figures are the textbook Simpson's paradox, reversing once split by department?

UC Berkeley's overall figures showed men admitted at a clearly higher rate, but the gap vanished department by department: women had applied in larger numbers to the most competitive programmes, which rejected everyone more often.

Which Scientific American columnist's 1959 'three prisoners' puzzle is Monty Hall with a pardon instead of a car?

Martin Gardner's puzzle has a warden name one of two condemned cellmates; the asker's own odds do not improve at all, while the unnamed prisoner's do — a result readers fought just as hard three decades later.

Which city names the paradox of a coin-flip game whose payout doubles each round, giving an infinite expected value?

The St. Petersburg paradox takes its name from the academy journal in which its 1738 analysis appeared, arguing that people value money by its usefulness rather than its face amount — the birth of the idea of diminishing utility.

What is the chance of being dealt a five-card royal flush, ten to ace in one suit, from a 52-card deck?

A royal flush comes up 1 in 649,740 deals, so a player dealt 50 hands an hour would wait about 18 months of continuous play to see one on average.

How many six-number combinations can a 6/49 lottery draw — the tickets needed to guarantee the jackpot?

There are 13,983,816 combinations, the result of 49 × 48 × 47 × 46 × 45 × 44 divided by the 720 orderings of six numbers, so a single line faces odds of about 1 in 14 million.

Under Benford's law, roughly what share of numbers in real-world data such as street addresses begin with the digit 1?

About 30% of leading digits are 1 and fewer than 5% are 9, a skew so reliable that forensic accountants use it to flag invented figures in tax returns and expense claims.

Whose 'law' holds that an ordinary person should expect a one-in-a-million event — a 'miracle' — about once a month?

J. E. Littlewood, the Cambridge analyst, reckoned we notice about one event a second while awake — a million a month — so a one-in-a-million coincidence is due monthly and nothing supernatural is needed to explain it.

Which betting system, popular in 18th-century France, doubles the stake after each loss so one win recovers everything?

The Martingale is certain to win only with infinite money and no table limit; in practice a run of six losses at even-money roulette odds, about a 2% event per sequence, wipes out 63 units and the plan collapses.

Which constant did the Comte de Buffon show can be estimated by dropping needles onto a floor of parallel planks?

The chance that a needle as long as the plank width crosses a line works out to 2/π, so counting crossings estimates π — one of the earliest problems in geometric probability, posed in the 18th century.

Which French mathematician posed the 1889 'box paradox' of gold and silver coins, identical in structure to Monty Hall?

Joseph Bertrand wrote the puzzle into his Calcul des Probabilités to show that merely counting cases can mislead: most people answer one half for the coin left in the drawer, and most are wrong.