Mark Hebner

Carl Friedrich Gauss was born in 1777 in Brunswick, Germany, the son of a bricklayer. His mathematical gifts were apparent almost from infancy. The stories are famous: as a young schoolboy, asked by his teacher to sum all the integers from 1 to 100, he produced the answer almost instantly — recognizing that the numbers form 50 pairs each summing to 101, giving 5,050. ifa.com/videos/pioneers-of-p…

Adrien-Marie Legendre is our 13th Pioneer of Probability. In 1801, Giuseppe Piazzi found a new world — then lost it. All he left other astronomers were a handful of imperfect observations. The question was not “do we have data?” It was “given noisy, inconsistent data, what is the best estimate of the true orbit?” In 1805, Adrien-Marie Legendre answered in four pages: minimize the sum of squared errors. That is least squares. It became the foundation of regression — and of the factor models, betas, and performance attribution investors use today. Gauss later claimed he had the method first. The tool that actually spread through science was Legendre’s. The 13th of 18 Pioneers of Probability episodes. lnkd.in/gjTkJkcm #PioneersOfProbability #LeastSquares #FactorInvesting #IndexInvesting

Welcome to this 12th of 18 Pioneers of Probability time travel videos. Pierre-Simon Laplace was born in 1749 in Normandy, the son of a peasant farmer. His Théorie Analytique des Probabilités, published in 1812, was the first comprehensive mathematical treatise on probability. And his Philosophical Essay on Probabilities — written as a non-mathematical introduction to the same ideas — remains one of the most lucid expositions of probabilistic thinking ever written. In that essay, Laplace introduced what became known as Laplace's demon — a thought experiment about determinism. If an intellect could know the position and momentum of every particle in the universe, he wrote, nothing would be uncertain. Probability, in Laplace's view, was not a property of the world. It was a measure of our ignorance about the world. A rational tool for making decisions when knowledge is incomplete. That philosophical position — probability as a measure of ignorance rather than a feature of reality — is one of the deepest ideas in the history of science. It is still debated today. Laplace extended de Moivre's normal approximation, pushing this work toward what would later become the Central Limit Theorem. He applied probability to astronomy, showing how to estimate the true position of a celestial body from a series of imperfect measurements. He developed generating functions — an important tool later used widely in probability and statistics. He also made probability respectable. Before Laplace, probability was associated with gambling, legal disputes, and philosophical puzzles. After Laplace, it was a branch of mathematics as rigorous as calculus — with formal definitions, proved theorems, and systematic methods of application. Every field that uses statistics today — medicine, economics, physics, engineering, social science — builds on foundations that Laplace helped establish. For investors, Laplace's classical definition carries a direct and important implication today. See the other Pioneers of Probability videos here: ifa.com/coins

Here is a game. We flip a fair coin, repeatedly, until it lands tails. If tails appears on the first flip, you win two dollars. If it first appears on the second flip, four dollars. The third, eight dollars. Each additional flip doubles the prize. To play this game, you have to pay a fee upfront: Your stake. How much would you pay to play? Take a moment. Think about it. The expected value of this game — the probability-weighted average of all possible payouts — is infinite. Literally infinite. By the logic of every probability theorist from Huygens to de Moivre, you should be willing to pay any finite sum to play a game with an infinite expected return. Would you pay a thousand dollars? Ten thousand? Everything you own? Almost nobody would. And that gap — between what the mathematics said you should do and what every rational person actually does — was one of the most important puzzles in probability theory for the first half of the eighteenth century, until Daniel Bernoulli found the answer. Welcome to Pioneers of Probability with me, Mark Hebner. Daniel Bernoulli worked here as a chair in mathematics at the Imperial Academy of Sciences in St Petersburg from 1725 to 1733. The puzzle was originally posed in 1713 by Nicolaus Bernoulli — Daniel's cousin — in a letter to the French mathematician Pierre Rémond de Montmort.

It's London, 1733. Picture one of the greatest mathematicians of his generation — a personal friend of Isaac Newton, a fellow of the Royal Society, a man Newton himself would redirect questioners to, saying: go to Mr. de Moivre, he knows these things better than I do "Sir Isaac, I don't understand how your calculus calculates a planet's speed at this point in its orbit. Could you show me the proof?" "Go to Mr De Moivre, he knows things better than I do these days" Now picture that man spending his days in a London coffee house, solving gambling problems for strangers in exchange for shillings. That was Abraham de Moivre. A French refugee who fled religious persecution in 1688, spent sixty years in England, "How can I help you?" and never once obtained a university position — not because he lacked ability, but because he was foreign and Protestant in a world that rewarded neither. He also discovered one of the most influential curves in the history of science. Welcome to the Pioneers of Probability with me Mark Hebner In the last episode, Yakob Bernoulli proved the Law of Large Numbers. Repeat a trial enough times, and the observed frequency converges toward the true probability. Repetition reveals truth. But Bernoulli's theorem left a question unanswered — one that any intellectually intelligent person would immediately want to ask.

Flip a coin. Heads or tails? You know going in that the odds of heads is one and two. In episode two, Cardano would have told you that. In three, Galileo would have counted it. In four and five, Pascal and Fermat would have used it to divide a gambling pot. In six, Huygens would have calculated your expected winnings. But here's a different question, one nobody had thought to ask formally until a Swiss mathematician did. The Bernoulli family is one of the most remarkable dynasties in the history of mathematics. Jacob was the eldest of the brothers who made the family famous. His brother Johann was equally brilliant. The two of them feuded bitterly for much of their lives, competing over priority, prestige, and problems. and Jacob's nephew Daniel will appear later in this series extending the family's contributions into economics and utility theory. But Jacob was first and the question he spent his career answering was in some ways the most important one in the whole story we are telling. Everything built before Bernoulli started from a known probability, a fair die, a fair coin, a fair game. The mathematics was about calculating what follows from that starting point. But the real world almost never gives you the starting point. You don't know whether a coin is fair until you flip it many times. You don't know the true return of the stock market until you observe it across many years. You don't know whether a medical treatment works until you run many trials. Let the virtue of the stone cleanse the poison. In the real world, probability is not given. It is estimated from data with uncertainty. In 1689, using a metaphorical urn filled with an unknown amount of black and white pebbles, Jacob Bernoulli was the first to rigorously prove a law showing that this estimation can be reliable. He showed repetition really does reveal truth. He called it the golden theorem. We call it the law of large numbers.

You already know two things about Gottfried Wilhelm Leibniz, born in Germany in 1646 — even if you don't know you know them. The first: he co-invented calculus in 1676, simultaneously and independently of Isaac Newton, igniting one of the most bitter priority disputes in the history of science. The second: he believed we live in the best of all possible worlds. A claim so optimistic that French Enlightenment Philosopher & Satirist Voltaire spent an entire novel mocking it. But here is what that optimism actually meant to Leibniz: this is the best world — and reason, logic, and probability are the tools we use to navigate it more wisely. His faith in the world and his faith in rational method were the same faith. But there is a third Leibniz that history has largely overlooked. A legal theorist. A philosopher of evidence. One of the first to argue systematically that probability wasn't just a tool for counting dice — it was a language for measuring belief itself. When you roll a die, you can run the experiment a thousand times and watch the frequencies emerge. The mathematics we have seen in previous episodes from Cardano, Galileo, Pascal, Fermat, and Huygens built was designed for exactly that: repeatable events with countable outcomes. But what about a murder trial? You cannot run a murder trial a thousand times and observe the frequency of guilt. The event happened once. The evidence is partial. The witnesses are unreliable. And yet a judge must still reach a verdict — must still make a decision about what probably happened, based on incomplete information. Leibniz argued that this kind of reasoning — legal reasoning, historical reasoning, reasoning about any singular unrepeatable event — could, in principle, be given a mathematical structure. And he spent decades trying to build it. Welcome to Pioneers of Probability. Leibniz trained as a lawyer before he became a mathematician. He spent years working as a diplomat and legal advisor, navigating the messy human business of courts, contracts, testimony, and judgment. And in that world — the world of law rather than the world of games — he encountered a kind of uncertainty that the existing mathematics of probability couldn't handle. His key insight was to reframe what probability actually measures.

If you could write one rule to make every gamble fair, what would it be? By 1655, word had spread through the mathematical circles of Paris that something remarkable had happened the previous summer. Two mathematicians — Pascal and Fermat — had exchanged a series of letters that solved a centuries-old gambling problem and, in doing so, helped establish a new field of study. A young Dutch physicist visiting the city heard the rumors. He tried to get hold of the letters. He couldn't. So Christiaan Huygens did what any serious mathematician would do. He worked it out for himself. Welcome to Pioneers of Probability. You may know Huygens from a different story entirely. He invented the pendulum clock — at the time, the most accurate widely used timekeeping device. He discovered Titan, Saturn's largest moon. He developed the wave theory of light that would later challenge Newton. By any measure, he was one of the great scientific minds of the seventeenth century. And yet the contribution that earns him a place in this series was, in some sense, a side project — a short book he wrote in his late twenties, almost as an exercise. It became one of the most influential texts on probability of its century.

It's the summer of 1654. Two mathematicians in France are exchanging letters about a gambling problem. One is Blaise Pascal — young, brilliant, already famous. The other is Pierre de Fermat — a magistrate in Toulouse, a lawyer by profession, a mathematician by pure obsession. He holds no university position. He publishes almost nothing. Mathematics is, technically, his hobby. Together, over a handful of letters written that summer, they will lay the foundations for a new branch of science. You may know Fermat's name from a different story — the margin note he left in a copy of Arithmetica, claiming a proof he never wrote down, that kept mathematicians searching for 350 years. That story is real. But it isn't this one. This one is about a question that had frustrated gamblers, merchants, and mathematicians for centuries before Fermat sat down to answer it. Welcome to Pioneers of Probability. The problem is called the Problem of Points — and it is deceptively simple. Two players are gambling. The game is fair — each has an equal chance of winning each round. They agree to play until one of them wins a set number of points. Then, for whatever reason, they have to stop early. Let's imagine the score is 9-1 and the winning score is 10. How do you divide the pot? It sounds like an accounting question. It isn't. It's a question about the future — about how to put a number on something that hasn't happened yet. And for centuries, no satisfactory solution existed.

It's Florence, around 1620. The grand duke's courtiers are arguing — again — about dice. Specifically, why a total of ten seems to come up more often than a total of nine. It makes no sense to them. Both totals can be made from exactly six combinations of three numbers. Six and six. Equal. So why does ten keep winning? They bring the question to the one of the most renowned thinkers  in all of Italy — Galileo Galilei. Yes, that Galileo. The one with the telescope. But today he's not looking at the stars. He's looking at dice. And what he finds will reveal something uncomfortable about the way all of us think about chance. Welcome to Pioneers of Probability. Before Galileo, people judged odds by instinct. They played, they watched, they developed hunches. Experienced gamblers noticed patterns — ten beats nine over thousands of throws — but they couldn't explain why. The numbers looked equal. Their experience said otherwise. And without a method to settle the question, argument was all they had. The problem wasn't the dice. It was the assumption that equal combinations mean equal chances. Nobody had thought to count every single way three dice can actually land. Galileo did.

It's the year 1200. You're a merchant in Pisa — one of the busiest ports in the Mediterranean. You need to split a shipment of pepper between three buyers, convert Florentine coins to Genoese pounds, and calculate your profit before the tide turns. Your tools? Roman numerals. Try multiplying XLVII by CCIX. Good luck. Before Fibonacci, Europe was stuck. Roman numerals worked fine for carving dates into marble. But for commerce — for the messy, high-stakes arithmetic of medieval trade — they were nearly useless. There was no zero. No place value. Multiplying large numbers required a physical abacus and years of training. Meanwhile, across the Mediterranean, Arab merchants were calculating faster, more accurately, and with less effort than anyone in Europe. Their secret was ten symbols — the digits 0 through 9 — arranged in a place-value system borrowed from Indian mathematicians centuries earlier. With those ten digits, any number could be written, added, multiplied, divided. No abacus required. Numbers became fluid. Calculation became fast. And one Italian teenager was watching.

It's 1550, and a physician in Milan has a problem. He's brilliant — one of the most famous doctors in Europe. He's also broke. Again. Because Girolamo Cardano cannot stop gambling. Cards, dice, chess — he plays them all, almost every day, for decades. He loses the family estate. He pawns his wife's jewelry. And somewhere between the bad bets and the borrowed money, he starts to notice something. The dice aren't random. Not really. They follow rules.

IFA.com - Pioneers of Probability: Leonardo Fibonacci youtu.be/jE_BGlfXPQk?is=ebiE…

It's the summer of 1654. Two mathematicians in France are exchanging letters about a gambling problem. One is Blaise Pascal — young, brilliant, already famous. The other is Pierre de Fermat — a magistrate in Toulouse, a lawyer by profession, a mathematician by pure obsession. He holds no university position. He publishes almost nothing. Mathematics is, technically, his hobby. Together, over a handful of letters written that summer, they will lay the foundations for a new branch of science. youtu.be/BVSBlUkdRTA?si=TWI6…

IFA.com - Pioneers of Probability: Christiaan Huygens youtu.be/ShIrT8HgMAU?si=UEGm…