Probability

The golden theorem: the law of large numbers

Even the most foolish of men, by some instinct of nature, by himself and without any instruction, is convinced that the more observations have been made, the less danger there is of missing the mark. — Jakob Bernoulli, Ars conjectandi (1713), part IV With a die, we know the chance of each face in advance: $\frac16$. But what is the chance that a man of 40 will live another twenty years? That it will rain tomorrow? That a medicine will work? In these cases there are no equal faces to count. Jakob proposed another route: observe many times and use the frequency as an estimate of the probability. If, of many men aged 40, half reached 60, the chance must be close to one half.

October 11, 2026 Read

The mystic hexagram: six points on a conic

In February 1640 a sheet printed on one side only, like a poster, circulated in Paris under the title Essay on conics. It had no proofs, only definitions, three lemmas and a list of results. The author was 16 years old and his name was Blaise Pascal. When Father Mersenne showed the sheet to Descartes, the reaction was suspicion: Descartes thought the young man had taken everything from Girard Desargues, the architect and geometer Pascal admired. Desargues’s influence was real, and Pascal himself acknowledged it. But the central result on the sheet was new.

October 8, 2026 Read

Fermat and Pascal: fairly dividing an interrupted game

Two players bet the same amount on a game of several rounds: whoever first wins a certain number of rounds takes everything. Midway through, the game has to be stopped. How should the money be divided fairly? This is the problem of points. It looks simple, but it defeated mathematicians for more than 150 years. The wrong answer In 1494 the Italian friar Luca Pacioli proposed dividing the prize in proportion to the rounds already won. If A has won 2 rounds and B has won 1, A takes $\frac23$ of the prize.

October 7, 2026 Read

Classical probability and the Gaussian integral

Mathematical probability began with games of chance, in the letters between Pascal and Fermat in 1654, and gained its first great theorem with Jakob Bernoulli’s law of large numbers. Laplace turned it into a complete theory. His first work on the subject dates from 1774; the Analytic Theory of Probabilities, of 1812, gathered almost forty years of research, and the Philosophical Essay on Probabilities, of 1814, explained it all to the public without formulas.

October 30, 2026 Read

The witch in statistics: the Cauchy distribution

Agnesi died in 1799 without imagining that her curve would have a second life, far from geometry. In the nineteenth century it reappeared in probability theory, as one of the most treacherous examples in statistics. A probability density We saw in The area under the curve that the total area under $y = \frac{1}{1 + x^2}$ is $\pi$. Dividing by $\pi$, the area becomes 1, and the curve becomes a probability density:

October 29, 2026 Read

The Monte Carlo method: calculating by rolling dice

In 1946 the Polish mathematician Stanisław Ulam, von Neumann’s colleague at Los Alamos, was recovering from encephalitis and passing the time playing solitaire. He wondered what the probability was that a game would come out. Working it out by combinatorics was hopelessly complicated. But there was another way: play hundreds of games and count how many came out. With a computer, one could play thousands. Ulam told von Neumann the idea, and he saw its reach at once: the same trick served for the problem occupying them at Los Alamos, the path of neutrons in a bomb or a reactor, where each neutron is deflected, absorbed or multiplied at random. Instead of solving the equations, simulate thousands of neutrons, drawing lots for what happens to each one, and take averages. Nicholas Metropolis named the method after the Monte Carlo casino, where an uncle of Ulam’s used to gamble.

October 27, 2026 Read

Banburismus: weighing evidence in decibans

The bombe was fast, but not fast enough to test every rotor order of the naval Enigma. In 1940 Turing invented a pencil-and-paper method to rule out most of them before switching on the machines. Because the sheets used in the method were printed in the town of Banbury, his colleagues named it Banburismus. The method is statistical. And to devise it, Turing reinvented one of the central ideas of modern statistics.

October 18, 2026 Read

The Bernoulli numbers: adding a thousand powers in seven minutes

With the help of this table it took me less than half of a quarter of an hour to find that the tenth powers of the first thousand numbers, added together, give 91,409,924,241,424,243,424,241,924,242,500. — Jakob Bernoulli, Ars conjectandi (1713), part II The story goes that young Gauss added the numbers from 1 to 100 in seconds, pairing the first with the last: $1+2+\cdots+n=\frac{n(n+1)}2$. For squares and cubes there are formulas too, known since Archimedes and the Arab mathematicians:

October 11, 2026 Read

The cycloid: the toothache curve

Paint a dot on a bicycle tyre and push the bike in a straight line. The dot rises, makes an arch and comes back down to touch the ground, once for every turn of the wheel. The curve it traces is called the cycloid. In the 17th century it was the fashionable curve. The French called it the roulette. So many mathematicians quarrelled over discoveries about it that it earned a nickname: the Helen of geometers, after Helen of Troy, who started a war.

October 8, 2026 Read

The central limit theorem

Roll one die: each face has the same chance, and the graph of the probabilities is flat. Roll two and add: the sum 7 is the most likely, and the graph becomes a triangle. Roll ten, thirty, a hundred: the graph of the sum gradually takes the shape of a bell. The same happens if, instead of dice, we add coins, roulette wheels or measurement errors. This surprising fact is the central limit theorem.

October 30, 2026 Read

The biased coin that becomes fair

Suppose you have a crooked coin. It lands heads with a probability $p$ that you do not know, perhaps $0.7$, perhaps $0.55$. Can you use it for a perfectly fair, fifty-fifty draw? In the same 1951 paper in which he spoke of the “state of sin” of those who manufacture random numbers, von Neumann gave a short and surprising answer. The trick Toss the coin twice: if it comes up heads, tails, say “heads”; if it comes up tails, heads, say “tails”; if it comes up heads, heads or tails, tails, forget it and toss twice more. Why is it fair? The two tosses are independent, so

October 27, 2026 Read

The legacy of James Bernoulli

James Bernoulli lived only fifty years and published little in his lifetime, almost all of it in short papers in the Acta Eruditorum. Even so, few mathematicians left so many words, symbols and ideas that we still use every day: integral, Bernoulli trial, binomial distribution, Bernoulli numbers, law of large numbers, stochastic. The word integral Leibniz called the operation inverse to the derivative calculus summatorius, the calculus of sums, and wrote it with $\int$, an elongated S for summa. In May 1690, solving the isochrone problem, Jakob wrote that the two expressions had “equal integrals”. It was the first time the word integral appeared in print in a mathematical text. Johann adopted the term, and in 1696 Leibniz agreed to swap his “summatory calculus” for the Bernoullis’ integral calculus. The symbol stayed Leibniz’s; the name, Jakob’s.

October 11, 2026 Read