Pioneers of Probability · Episode 10

This video is for informational and educational purposes only and does not constitute a solicitation or recommendation to buy or sell any security. The historical and mathematical concepts discussed are intended to illustrate the development of probability theory and its relevance to investing. Past performance is not indicative of future results. All investing involves risk, including the possible loss of principal. The examples provided are hypothetical and based on historical index data, not an actual investment. Index returns do not reflect the performance of any actual portfolio or the deduction of advisory fees. Content is AI assisted.

Daniel Bernoulli, the St. Petersburg Paradox, and why a dollar is not always worth a dollar

◆ The Question

In the spring of 1725, a letter arrived in Basel bearing the seal of Empress Catherine I of Russia. It was an invitation to join the newly founded Imperial Academy of Sciences in St. Petersburg — one of the most prestigious scientific appointments in Europe. The letter was addressed to Daniel Bernoulli, twenty-five years old, who had just completed a medical degree he never wanted, in a field he had studied only because his father had forbidden him to pursue mathematics.

The father in question was Johann Bernoulli, chair of mathematics at the University of Basel and one of the most formidable mathematical minds of the early eighteenth century. Johann had grown up watching his own older brother Jacob turn mathematics into a professional empire, and when Jacob died of tuberculosis in 1705, Johann moved into his chair without hesitation. He was brilliant, ambitious, and extraordinarily competitive — a combination that made him a difficult father. He had pushed Daniel toward business, then toward medicine, and had only relented enough to teach the boy calculus privately, in stolen hours, while Daniel was supposed to be studying anatomy.

Daniel accepted the Russian invitation. His older brother Nicolaus came with him — the Empress had agreed to fund a second position to secure Daniel's acceptance. Within eight months of arriving in St. Petersburg, Nicolaus was dead of fever. Daniel was left alone in a city he found cold, harsh, and deeply uncongenial, in a country where he did not speak the language, with a brother buried in foreign soil.

His father sent one of his own students — a twenty-year-old Swiss mathematician named Leonhard Euler — to assist Daniel in his research. The collaboration between Bernoulli and Euler would become one of the most productive partnerships in the history of mathematical physics. Daniel stayed in St. Petersburg for eight years.

◆ The Insight

It was during those St. Petersburg years that Daniel began working on the problem that would define his contribution to probability theory. The problem had been posed in a letter by his cousin Nicolaus I Bernoulli in 1713 and had been circulating, unresolved, among Europe's mathematicians ever since. It came to be known as the St. Petersburg Paradox, named for the city and the journal — the Papers of the Imperial Academy of Sciences in Petersburg — in which Daniel's solution was eventually published in 1738.

The paradox works as follows. A casino flips a fair coin repeatedly until it lands tails. If tails appears on the first flip, the player wins 2 dollars. On the second flip, 4 dollars. The third, 8 dollars. Each additional flip doubles the prize. How much should a rational person pay to play this game?

The expected value calculation — multiplying each possible outcome by its probability and summing the results — gives an answer that had been paralysing mathematicians for twenty-five years: infinity. Literally infinite expected value. By the logic of every probability theory from Huygens onward, a rational agent should pay any finite sum to play a game with infinite expected return. Yet no reasonable person would pay even a hundred dollars. The mathematics said one thing. Every human instinct said something utterly different.

Daniel's insight was that the mathematics was not wrong. The assumption behind it was.

◆ The Proof

The hidden assumption was that money has fixed, linear value — that one hundred dollars is always worth exactly ten times ten dollars, regardless of how much money you already have. Daniel showed this is simply false. The value of money is not its quantity but its utility: the real-world benefit it provides to the person who receives it. And that utility diminishes as wealth increases.

Imagine you have nothing. A thousand dollars transforms your life. Now imagine you already have a million dollars. Another thousand is pleasant but changes nothing fundamental. The same dollar amount produces dramatically different real-world benefit depending on the recipient's starting position. This is diminishing marginal utility — and once you incorporate it into the calculation, the paradox dissolves.

The IFA MarketCoin® for Daniel Bernoulli encodes his solution directly on its reverse. The coin shows u(w) = ln(w) — logarithmic utility of wealth — alongside the expected utility formula EU = Σ(1/2)^n × ln(2^n), beneath the inscriptions "St. Petersburg Paradox Resolved," "Utility of Wealth," and "Expected Utility Theory." When you replace dollar amounts with their logarithmic utilities and recalculate the expected utility of the St. Petersburg game, the infinite value collapses into a finite number — typically a modest one, consistent with what real people are actually willing to pay.

The implications extended far beyond the paradox. Daniel had demonstrated that a gamble with a fifty percent chance of doubling your wealth and a fifty percent chance of losing everything has an expected dollar gain of zero but a deeply negative expected utility. Losing everything destroys far more utility than doubling your wealth creates. The mathematics of ruin is categorically different from merely losing money. This is not timidity or irrationality. It is the correct mathematical response to the curvature of utility.

The 1738 paper, Specimen Theoriae Novae de Mensura Sortis — Exposition of a New Theory on the Measurement of Risk — was Daniel's masterwork in probability. It was published the same year as Hydrodynamica, his masterwork in physics, the book that founded the science of fluid mechanics and explained, in principle, why airplane wings generate lift. Both appeared in 1738. His father Johann responded by publishing a book called Hydraulica, backdating it to appear to predate Hydrodynamica and stealing credit for Daniel's discoveries. In 1734, when he and Daniel had shared a prize from the Paris Academy, Johann had been so furious that his son was considered his equal that he banned Daniel from his house.

Daniel had put on the frontispiece of Hydrodynamica: "Daniel Bernoulli, Son of Johann." It was a gesture of deference. His father responded with plagiarism. There is no evidence Daniel ever fully recovered from it.

◆ The Legacy

Between 1725 and 1749, Daniel won ten prizes from the Paris Academy of Sciences, covering work in astronomy, gravity, tides, magnetism, ocean currents, and the behavior of ships at sea. He won a chair of physics at Basel in 1750 and taught there for twenty-six years, becoming one of the most admired scientists in Europe. He outlived his father by two decades and died in Basel in 1782 at the age of eighty-two.

The expected utility framework he built became the foundation of modern decision theory. Kahneman and Tversky's Prospect Theory, which won the Nobel Prize in Economics in 2002, can be understood in part as a detailed map of exactly how and where human utility judgments deviate from Daniel's logarithmic model. The Kelly Criterion, used by professional investors and quantitative traders to determine optimal bet sizing, is a direct expression of expected log-utility maximization. Harry Markowitz's mean-variance optimization framework — the basis of modern portfolio theory — is closely related to Bernoulli's framework under specific assumptions about return distributions.

◆ Your Money

Daniel Bernoulli's central insight is more relevant to modern investors than any piece of market analysis published this week. Utility is not linear. Losses hurt more than equivalent gains help. The difference between losing 50 percent of your portfolio and gaining 50 percent is not zero: losing half requires a subsequent 100 percent gain just to break even.

This asymmetry is why diversification is not merely prudent but mathematically necessary for any investor whose utility is even approximately logarithmic. A concentrated bet in a single stock introduces the kind of variance that destroys expected log-utility without improving expected return. The expected value of a concentrated position may equal or exceed the expected value of a diversified one — but the expected utility is lower, because the downside scenarios are far more damaging than the upside scenarios are beneficial.

The rational response — the one Bernoulli's mathematics demands — is to hold the whole market, minimize uncompensated risk, keep costs low, and hold long enough for the underlying expected return to reveal itself. Not because this is conservative. Because diminishing marginal utility is a mathematical fact about how human beings actually value wealth, and a portfolio strategy that ignores it is optimizing the wrong thing.

Daniel Bernoulli replaced the infinite expected value of the St. Petersburg game with a finite, honest number. Every rational investor should do the same for their own portfolio: replace the fantasy of maximum expected return with the mathematically correct goal of maximizing expected utility. They are not the same thing. Bernoulli proved it in 1738.

Sources: Bernoulli, D. (1954). Exposition of a new theory on the measurement of risk. Econometrica, 22(1), 23–36. (Original work published 1738.) Stigler, S. M. (1986). The history of statistics. Harvard University Press. MacTutor History of Mathematics, University of St Andrews.

Disclosure: This article is for informational and educational purposes only and does not constitute a solicitation or recommendation to buy or sell any security. Past performance is not indicative of future results. All investing involves risk, including the possible loss of principal. Any historical return examples, if referenced, are hypothetical illustrations based on published index data and are not reflective of actual investor experience. Any discussion of diversification, expected utility, portfolio construction, or investment strategy reflects general educational concepts and should not be interpreted as individualized investment advice.

Content is AI-assisted. Index Fund Advisors, Inc. is a registered investment adviser. For additional information, please visit adviserinfo.sec.gov or www.ifa.com.

The Exile at Slaughter's Coffee House | Abraham de Moivre (PoP EP9)

Claude Hebner | IFA.com
Tuesday, August 11, 2026

Pioneers of Probability · Episode 9

Abraham de Moivre, the normal curve, and the shape that appears wherever chance accumulates

◆ The Question

In the autumn of 1685, a teenage boy sat in a French prison because he attended the wrong church.

Abraham de Moivre was eighteen years old when Louis XIV revoked the Edict of Nantes, the law that had guaranteed French Protestants the right to practice their faith. The revocation triggered an immediate wave of persecution. Huguenot churches were demolished. Protestant schools were closed. Those who refused to convert were arrested. De Moivre, whose father was a Protestant surgeon in Vitry-le-François in the Champagne region, was imprisoned. He read what mathematics he could find in his cell. According to one account, pages of Newton's just-published Principia Mathematica were smuggled to him there, and he read them by whatever light was available.

When he was released, he left France immediately and permanently, arriving in London as a refugee in 1687. He would live in England for the remaining sixty-seven years of his life, becoming a Fellow of the Royal Society, a personal friend of Isaac Newton and Edmond Halley, one of the most accomplished probabilists of his century — and never, despite every effort and qualification, obtaining a university position.

That last detail matters. It shaped everything about de Moivre's life and work. To earn a living, he became a private mathematics tutor, visiting students' homes and conducting sessions in the coffee houses of London, particularly in Slaughter's Coffee House on St. Martin's Lane, where gamblers would seek him out and pay him shillings to calculate odds. Newton himself, when asked mathematical questions he considered beneath his current attention, would reportedly tell visitors: "Go to Mr. de Moivre; he knows these things better than I do." The greatest mathematician alive was directing students to a man who made his living in a coffee house. De Moivre died in poverty at eighty-seven, despite that endorsement.

◆ The Insight

Jacob Bernoulli had proved, in Ars Conjectandi, that the proportion of successes in repeated trials converges toward the true probability. This was a profound result — but it raised an immediate practical question that Bernoulli had not fully answered: how does the convergence happen? How quickly? And while you are accumulating trials — while you are still in the noisy middle of the process — how are your results distributed around the true value? How often do you land close? How often far away?

De Moivre spent years on this question. He was extending Bernoulli's work on the binomial distribution — the probability of getting exactly k successes in n trials when each trial has probability p of success. For small n, this is manageable. For large n, the calculation becomes astronomically complex. De Moivre asked: as n grows very large, what shape does the binomial distribution approach?

The answer he found was a curve that would eventually be called the normal distribution. The bell curve. The shape that emerges wherever many small independent influences add together to produce an outcome. And what astonished de Moivre — and what has astonished every statistician since — is how universal it is. It appears in the heights of a population, in the measurement errors of astronomers, in the scores of students on an exam, in the returns of financial assets over time. The same shape, drawn from completely different processes, arising from completely different forces. Order emerging from the accumulated chaos of independent events.

◆ The Proof

De Moivre first published his approximation in a privately circulated Latin pamphlet in 1733 — characteristically modest, circulated among a few trusted colleagues rather than published broadly. He later incorporated it into the second edition of his masterwork, The Doctrine of Chances, in 1738, and the third edition in 1756. The core result is the De Moivre-Laplace theorem, displayed on the reverse of the IFA MarketCoin® for de Moivre alongside a histogram transitioning smoothly into a bell curve, beneath the inscriptions "Discoverer of the Normal Curve" and "De Moivre-Laplace Theorem." The formula states that the binomial distribution with parameters n and p is approximately normally distributed with mean np and variance np(1 − p) — written on the coin as Binomial(n, p) ≈ N(np, npq).

The result is not merely a computational convenience. It is a deep statement about why averages behave predictably even when the underlying data does not. When you take the average of many independent random variables — each following its own distribution, which might be skewed or lumpy or anything at all — the average tends toward a normal distribution. This is the seed of what would later be called the Central Limit Theorem, and de Moivre was the first to conjecture it explicitly, if not yet in its most general form. Laplace would later generalize and rigorously prove it. But the shape was de Moivre's discovery, seen first by a French exile doing calculations for gamblers in a London coffee house.

The Doctrine of Chances also contained the first clear statement of statistical independence — the principle that the probability of a compound event composed of independent sub-events is the product of their individual probabilities. This is one of the most frequently used tools in all of probability and statistics. It was stated explicitly by de Moivre and has been used by every probability textbook since.

De Moivre also made major contributions entirely outside probability. His formula in complex number theory — (cosθ i·sinθ)^n = cos(nθ) i·sin(nθ) — is taught in every undergraduate mathematics course. He was also, less famously, the first to discover Binet's formula, the closed-form expression connecting the nth Fibonacci number to the Golden Ratio. The man solving gambling problems at Slaughter's was, in his spare moments, spanning the entire width of mathematical knowledge.

◆ The Legacy

De Moivre lived to eighty-seven — remarkable longevity for any era. Near the end of his life, according to a story that circulated soon after his death, he noticed he was sleeping fifteen minutes more each night than the night before. He applied his professional habit: he calculated. If the increment held, his required sleep would reach twenty-four hours on November 27, 1754. He predicted he would die on that date. He did, reportedly in his sleep, having performed one final probability calculation on himself.

Johann Bernoulli — with whom de Moivre had a difficult relationship, connected partly through his brother Jacob and partly through overlapping mathematical territory — had once begged Leibniz to help de Moivre secure an academic position in Germany. Leibniz tried and failed. Newton and Halley tried in England and failed. De Moivre remained a private tutor for sixty years, the resident statistician of a coffee house, writing books that transformed the field from a marginalized appendage of gambling into the foundation of scientific inference. The normal curve now has Gauss's name on it. The theorem that connects it to the binomial carries de Moivre's. That division of credit seems, on balance, about right.

◆ Your Money

The normal distribution is the foundation of modern quantitative finance — and, in equal measure, its most consequential limitation.

Standard deviation, the primary measure of portfolio volatility, is a concept that only makes full sense within the framework de Moivre built. A portfolio with an annual standard deviation of 15 percent is one where, under the normal approximation, roughly two-thirds of annual returns fall within 15 percentage points of the mean, and roughly 95 percent fall within 30 percentage points. These are exact results for a normal distribution, and useful approximations for many real-world asset classes over typical market conditions.

That framework has underpinned nearly every advance in quantitative finance since Markowitz: portfolio optimization, options pricing, value-at-risk models, factor analysis. All of it rests on the bell curve de Moivre described at Slaughter's Coffee House.

The caveat is equally important. Real market returns have fatter tails than the normal distribution predicts. Extreme events — crashes, crises, sudden dislocations — happen more often than the bell curve suggests. Every serious risk manager knows this. The financial crisis of 2008 produced daily losses that were extremely improbable under normal distributional assumptions — and produced them on multiple consecutive days. The model breaks down in the extremes, and the extremes are precisely where the damage is greatest.

But you cannot understand where the model breaks down until you understand the model. And understanding the model means understanding what de Moivre discovered: that the bell curve is the natural shape of accumulated uncertainty — what chance looks like, in the aggregate, when the noise begins to average out. Build portfolios that can survive what the bell curve says should almost never happen. But begin, as every serious practitioner must, with the curve itself. The one de Moivre drew in a coffee house, three centuries ago, while waiting for the next gambler to need his numbers.

Sources: de Moivre, A. (1756). The doctrine of chances (3rd ed.). A. Millar. Stigler, S. M. (1986). The history of statistics: The measurement of uncertainty before 1900. Harvard University Press. Britannica: Abraham de Moivre.

Disclosure: This article is for informational and educational purposes only and does not constitute a solicitation or recommendation to buy or sell any security. Past performance is not indicative of future results. All investing involves risk, including the possible loss of principal. Any historical return examples, if referenced, are hypothetical illustrations based on published index data and are not reflective of actual investor experience. Statements regarding statistical concepts, market behavior, and portfolio construction reflect educational commentary and should not be construed as forecasts or guarantees of future market outcomes.

Content is AI-assisted. Index Fund Advisors, Inc. is a registered investment adviser. For additional information, please visit adviserinfo.sec.gov or www.ifa.com.

About the pen name: "Claude Hebner" represents a collaboration between Mark Hebner, founder and CEO of Index Fund Advisors, Inc., and Claude, Anthropic's AI. The research, historical narrative, and investment analysis in each article are the result of that partnership, combining human editorial oversight with AI-assisted research and drafting.