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50 free Probability trivia questions with answers — trivia quiz, new questions added Aug 2026.
Probability is the branch of mathematics that most reliably makes clever people wrong. A room of two dozen strangers is more likely than not to hold a shared birthday, switching doors on a game show really does double your chances, and a streak of one colour at the roulette table tells you nothing at all about the next spin — yet each of those facts has been argued over in print by people with doctorates. These 50 questions cover the odds behind dice, cards, coins and lottery tickets; the classic puzzles and paradoxes; the correspondence, treatises and wartime coin-tossing that built the subject from the 1650s onward; and the places chance turns up in daily life, from rain forecasts and hole-in-one insurance to a miscarriage of justice built on a bad statistic. Every answer comes with a short explanation of the why, not just the what. It suits a maths classroom starter, a pub-quiz round with a sting in it, or anyone who wants to find out whether their intuition about chance is as good as they think.
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Q 01How many people in a room are needed before a shared birthday between two of them is more likely than not?
23
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.
Q 02In the Monty Hall problem, what chance of winning the car does switching give once the host reveals a goat?
2/3
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.
Q 03What do statisticians call the belief that a fair coin, after ten heads in a row, is 'due' to land tails?
Gambler's fallacy
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.
Q 04How many riffle shuffles did Bayer and Diaconis show a 52-card deck needs before its order is effectively random?
7
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.
Q 05Roughly how many different orderings can a shuffled 52-card deck take, the number mathematicians write as 52 factorial?
About 8 × 10^67
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.
Q 06What are the jackpot odds for a Powerball ticket matching five white balls from 69 and the red ball from 26?
1 in 292,201,338
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.
Q 07Per Persi Diaconis's physics model, with what probability does a caught coin land the same way up it started?
About 51%
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.
Q 08Which physicist told Max Born in a 1926 letter that he was convinced God 'is not playing at dice'?
Albert Einstein
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.
Q 09How many ways can a 63-game March Madness bracket be filled in if every game is treated as a coin flip?
9.2 × 10^18
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.
Q 10In the secretary problem, what share of candidates do you reject before taking the first one better than all of them?
About 37%
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.
Q 11With two fair dice, what is the probability of a double six, the only way to throw a total of twelve?
1/36
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.
Q 12What is the probability that three fair coins all land heads, one of the two outcomes that void a three-way flip?
1/8
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.
Q 13What is the expected value of a single roll of a fair six-sided die, the average seen over thousands of throws?
3.5
Q 21Which Toulouse lawyer's 1654 letters with Blaise Pascal on a gambling problem founded probability theory?
Pierre de Fermat
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.
Q 22Which man of letters brought the 'problem of points', on splitting an unfinished game's pot, to Pascal in 1654?
Chevalier de Méré
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.
Q 23Which Italian polymath's Liber de ludo aleae, written around 1564, was the first systematic treatment of probability?
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.
Q 14If 1 driver in 1,000 is drunk and a breathalyser flags 5% of sober ones, how likely is a flagged driver drunk?
About 2%
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.
Q 15Which solicitor's 1999 murder conviction over her two sons' deaths hinged on a discredited '1 in 73 million' statistic?
Sally Clark
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.
Q 16Which sport did Gilovich, Vallone and Tversky study in the 1985 paper that called the 'hot hand' an illusion?
Basketball
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.
Q 17What house edge does a European single-zero roulette wheel carry, about half the American double-zero figure?
2.7%
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%.
Q 18At Monte Carlo in 1913, how many spins in a row came up black while bettors lost millions of francs backing red?
26
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.
Q 19Which game show, where players traded prizes for what hid behind numbered doors, gave the Monty Hall problem its name?
Let's Make a Deal
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.
Q 20In a normal bell curve, what share of values fall within one standard deviation either side of the mean?
68%
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.
Gerolamo Cardano
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.
Q 24Which Dutch scientist, famed for the pendulum clock, wrote the first published treatise on probability, in 1657?
Christiaan Huygens
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.
Q 25Which Swiss mathematician's posthumous Ars Conjectandi of 1713 proved the law of large numbers, his 'golden theorem'?
Jacob Bernoulli
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.
Q 26Which Huguenot exile in London described the bell curve in 1733 and wrote The Doctrine of Chances?
Abraham de Moivre
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.
Q 27Thomas Bayes, whose theorem updates a probability on new evidence, earned his living in what profession?
Presbyterian minister
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.
Q 28Which Welsh philosopher edited and published Bayes's essay in 1763, two years after its author's death?
Richard Price
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.
Q 29Which Soviet mathematician's 1933 Foundations of the Theory of Probability gave the subject its modern axioms?
Kolmogorov
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.
Q 30Which Los Alamos mathematician dreamed up the Monte Carlo method in 1946 while playing solitaire during an illness?
Stanisław Ulam
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.