Gambling and Probability in the 18th Century: The Classic Chicken-and-Egg Situation

Gambling and Probability in the 18th Century

Which influenced what? – That is the question!

In the 90-something decades of British Gambling, the Long Eighteenth Century is perhaps the most talked-about.

England was gripped by gambling fever,' writes British Historian Roy Porter in his 1982 book on English cultural history.

This was the time when people bet on practically ‘any future happenings‘ – from political events to births and deaths – and cards were dubbed ‘the opium of the polite.[1]

Rampant among the elite as well as those of the lower order, gambling upset the watchdogs of conduct, more so the gatekeepers of female morality.

For playful 18th-century writers, this was satire material of the highest order.[2]

Even the 1720 stock market crash—the South Sea Bubble that rocked England and ensnared the likes of Sir Isaac Newton—was convincingly explained by W. R. Scott, P. G. M. Dickson, and John Carswell using the ‘gambling mania' theory that was so well pronounced at that time.[3][4][5]

Now discarded by modern economic historians employing behavioural finance[6], this theory stands as a testament to how gambling was perceived in 18th-century Britain—as an evil that affected all levels of society.

How Did It Come to This?

While different theorists cite different reasons, one particular parallel drawn between gambling and the coeval ‘probability revolution' stands out.

Several pieces of English literature on gambling written between the early sixteenth and mid-eighteenth century extensively explore this connection.

An overwhelming preoccupation‘ was noted by statistician and actuarialist David Bellhouse with cheating at games of chance in these pieces.

In his article The Role of Roguery in the History of Probability[7], he highlights that cheating remains a constant discussion in gambling literature on that era, while the methods vary (think false dice, sleight of hand with cards and dice, perfect shuffles and card counting).

Towards the late seventeenth century, the text also gradually incorporates discussions on probability.

Now, two distinct perspectives review this literature in modern times: one advocating the accepted folklore that gambling birthed probability, and the other opining that probability influenced gambling by aiding in the formulation of strategies.

Bellhouse contended that while gamblers might have a grasp of probability, gambling isn't a catalyst strong enough to drive the advancements of probability theory.

This makes us wonder if the reverse is also true.

Understanding The Accepted Folklore: Probability Stemmed From Gambling

The idea that gambling birthed probability originates from historical facts that show how early probabilists had a penchant for studying games of chance.

Pay attention to this timeline:

  • Mid-13th century: a Latin poem called De vetula praises the numerical game Rithmomachia, references an early form of backgammon, and introduces the first known probability calculation for rolling three dice.[8]
  • 1494: Mathematician Luca Pacioli mentions the problem of points in his book Summa de arithmetica, geometrica, proportioni et proportionalità using the example of a fair game of chance.[9]
  • 1560s (published 1663): Italian polymath and habitual gambler Gerolamo Cardano (Jerome Cardan) writes Liber de Ludo Aleae (The Book on Games of Chance), aiming to calculate dice throw probabilities. He shows that odds are just the ratio of favourable to unfavourable outcomes, indicating that an event's probability equals the ratio of favourable outcomes to the total possible outcomes.10]
  • 1596 (published 1718): The great Galileo prepares a brochure called Soprale Scopere Dei Dadi (Findings on Dice Games) after solving the Duke of Tuscany's problem (1560) to explain the outcomes of Passedix, a game of chance played with three dice.[11]
  • 1654: Blaise Pascal and Pierre de Fermat co-create the mathematical theory of probability by solving the classical ‘problem of points' or ‘division of stakes' related to expected outcomes in a game of chance.[12]
  • 1657: Dutch mathematician and physicist Chistiaan Huygens writes Libellus de Ratiociniis in Ludo Aleae (Calculations in Games of Chance), the first book on mathematical probability based on games of chance (cards, dice, wagers, and lotteries).[13]
  • 1662: A book called La Logique ou l’Art de Penser is published anonymously in Paris (likely written by authors Antoine Arnauld and Pierre Nicole), using gambling to explain decision-making under uncertainty.[14]
  • 1684 and 1689 (published 1713): Jacob Bernoulli writes Ars Conjectandi (The Art of Conjecturing), containing the first version of the Law of Large Numbers (basis of Gambler's Fallacy) based on Christiaan Huygens' posed games of chance.[15]
  • 1713: Nicolaus Bernoulli introduces the St. Petersburg paradox based on the St. Petersburg game of coin flips in which he explains the general problem of doubling the bet in gambling.[16]
  • 1718: French mathematician Abraham de Moivre publishes the first textbook on probability theory, The Doctrine of Chances: A Method of Calculating the Probability of Event in Play, applying his theories to common gambling problems.[17]
  • 1754: Renowned mathematician Jean Le Rond d'Alembert mistakenly argues in his article Croix ou Pile (Cross or Tail) published in the monumental Encyclopédie that each coin flip landing tails increases the odds of heads on the next.[18]

As is evident, the first mathematical investigations into probability truly began only in the 16th century, picked up steam in the 17th century, and continued with equal gusto in the 18th century.

In short, probability showed up in math much later than in other areas!

Historians think it's because people in ancient times attributed randomness to the gods. Even the brainy Greeks, who loved math and logic, avoided studying it.

They saw random events like dice rolls as ways to talk to the gods, not things to figure out with math. Plus, math was all about certainty, so predicting uncertain events probably seemed weird.

Yet, this isn't a reasoning strong enough to address why gambling did not spur advancements in probability early on. With games of chance spanning back through human history, probability should have dated back similarly far.

If we assume as is commonly done, that probability theory owes its origin to gambling,' says author L.E. Maistrov, ‘it would be necessary to explain why gambling, which had been in existence for six thousand years, did not stimulate the development of probability theory until the seventeenth century, while in that particular century the theory originated on the basis of the same games of chance.‘[19]

According to him, the traditional connection between probability and gambling overlooks the subject of prehistory. Maistrov's opinion also aligns with German mathematician Ivo Schneider's.

The latter suggests that the 15th and 16th-century economy, influenced by the popularity of gambling and risk concerns, inspired the resolution of the problem of points[20].

Furthermore, while the timeline above suggests that the initial developments in probability theory may have been inspired by games of chance, we don't understand why it didn't continue in the 18th and 19th centuries as well.

A possible explanation could be the First Industrial Revolution that occurred between 1760 and 1840.

As industrialization progressed, probability theory found new applications, predicting the likelihood of equipment failures and other events.

The advent of quantum mechanics in the 1900s further boosted probability theory's significance, revealing the inherent uncertainty of events at the atomic level. For instance, scientists could now calculate the probability of radioactive decay of atoms.

Around this time, the insurance industry was emerging as well.

The aim was to collect more than what's paid out for accidents. Understanding odds was crucial to the companies, which led to actuaries advising on premium rates based on probability theory.

These new applications, in a way, liberated probability from its earlier associations with gambling and games of chance.[21]

Understanding The Counter Argument: Gambling Stemmed From Probability

Patrons of the second school of thought believe developments in probability theory birthed gambling strategies (and gambling as we know it today).

Coincidentally (or not), the first casino, Ridotto arrived in Italy in 1638[22], and by the 1700s commercial gambling houses were booming in Europe.

Interestingly, the period between the 16th and 18th century also saw a surge of new gambling games globally.

Here are some instances:

  • Ruff and Honours/Ruffe and Trump/Slamm in 16th and 17th-century England (derived from the 15th-century French game Triomphe)[23]
  • Piquet in 16th-century France[24]
  • Faro and Trente et Quarante (Thirty and Forty) or Rouge et Noir (Red and Black) in 17th-century France[25][26]
  • 21/Blackjack/Vingt-Un/Vingt-et-Un[27] and Roulette[28] in 18th-century France

Here's a source to play similar, but modern gambling games online for real money:

Now, it's unclear whether these advancements directly sprouted from the era's work on probability. But they do connect some dots.

For instance, the game of Roulette is a byproduct of Blaise Pascal's failed attempt at creating a perpetual motion machine, but we doubt if it has anything to do with probability.

Again, early gambling games offered players extremely favourable odds.

Yet, even after Johann and Jakob Bernoulli demonstrated the potential advantage of a small house edge with their Golden Theorem in 1713[29], operators wouldn't adopt it, viewing it as uncharted waters.

In his book The Gambling Century: Commercial Gaming in Britain from Restoration to Regency, Professor John Eglin from the University of Montana explains how it took government intervention to ban and regulate betting before operators realised the profitability of a small house edge.[30]

This suggests that development of the probability theory in the 18th century alone wasn't enough to shape gambling as we know it today.

Meanwhile, the first gambling strategies surfaced in the mid-eighteenth century, devised by the renowned English writer Edmond Hoyle.

A master hand at card game rules, Hoyle began teaching high society members the card game of whist in 1741, eventually publishing his rules in 1742.

These strategies, outlined in his book A Short Treatise on the Game of Whist[31], were based on simple probability principles.

Fun fact: The expression “according to Hoyle,” means “strictly according to the rules.” It entered common language as a testament to Edmond Hoyle's widely recognized expertise in the field. 

Yes, the timing of these occurrences aligns with advancements in probability theory, albeit in strange ways.

But correlation does not necessarily imply causation.

The Truth

In his book Gambling in Britain in the Long Eighteenth Century[32], Bob Harris challenges the prevalent narratives surrounding the gambling mania of that era itself.

He says the assumptions lack nuances and that “too many histories of gambling rely on impressionistic comment and rumour and gossip.”

Now, if this is the truth, the parallels drawn between probability and gambling, too, probably rest on general assumptions rather than concrete evidence.

Gambling didn't flourish in the 18th century just because games of chance were used to establish the probability theory.

Gambling didn't catalyse the probability theory either just because games of chance were used to establish it. It's the perfect chicken-and-egg problem to which there's no solution.


  1. Porter, R. (1982). English Society in the Eighteenth Century. Penguin Group.
  2. Evans, J.E. (2002). A sceane of uttmost vanity: The Spectacle of Gambling in Late Stuart Culture. Studies in Eighteenth Century Culture, 31(1), pp.1–20. Online
  3. Scott, W.R., Litt.D., & c. (1910). The Constitution and Finance of English, Scottish, and Irish Joint-Stock Companies to 1720. Cambridge: University Press. Online
  4. Dickson, P.G.M. (1968). The Financial Revolution in England: A Study in the Development of Public Credit 1688-1756. Routledge.
  5. Carswell, J. (1960). The South Sea Bubble. Cresset Press.
  6. Paul, H.J. (2011). The South Sea Bubble: An economic history of its origins and consequences. Routledge.
  7. Bellhouse, D. (1993). The Role of Roguery in the History of Probability. Institute of Mathematical Statistics.
  8. Bellhouse, D.R. (2000). De Vetula: A Medieval Manuscript Containing Probability Calculations. International Statistical Institute.
  9. Katz, V. J. (1993). A history of mathematics. HarperCollins College Publishers.
  10. Gorroochurn, P. (2012). Some Laws and Problems of Classical Probability and How Cardano Anticipated Them. Chance magazine. Online
  11. Lepri, G. (2021) Galileo Galilei e il suo contributo al calcolo delle probabilità: ‘Sopra le scoperte dei dadi’ lepri giulia. Online
  12. Ma, D. (2016). The problem of points. A Blog on Probability and Statistics. Online
  13. Huygens, C. (1657) Libellus de Ratiociniis in Ludo Aleae. Online
  14. Arnauld, A. and Nicole, P. (1662). La Logique ou l’Art de Penser. Translated by Baynes, T.S. Online
  15. Shafer, G. (1996). The significance of Jacob Bernoulli’s Ars Conjectandi for the philosophy of probability today. Journal of Econometrics, Elsevier. Online
  16. Peterson, M. (2019). The St. Petersburg Paradox. Stanford Encyclopedia of Philosophy. Online
  17. Moivre, A. de (1718). The Doctrine of Chances: A Method of Calculating the Probability of Events in Play. W. Pearson. Online
  18. Crépel, P.  (2013) L'article “Croix ou pile” du tome IV de l'Encyclopédie (1754). CNRS. Online
  19. Maistrov, L. E. (1974) Probability Theory: A Historical Sketch. New York: Academic Press.
  20. Schneider, I. (1988). The market place and games of chance in the fifteenth and sixteenth centuries. Mathematics from Manuscript to print. Ed. Cynthia Hay. Clarendon Press; New York : Oxford University Press.
  21. Karam, P.A. (2024). Eighteenth-Century Advances in Statistics and Probability Theory. Science and Its Times: Understanding the Social Significance of Scientific Discovery. Online
  22. McCoy, A.. (2024) A Basic History Of Casino Games. Ministry of Public Service – The Republic of Uganda. Online
  23. Cotton, C. (1674) The Compleat Gamestar. Princeton University. Online
  24. Parlett, D. Piquet: Historic Card Game Described by David Parlett. Online
  25. HistoryNet Staff. (2006). Faro: Favorite Gambling Game of the Frontier. Online
  26. Parlett, D. (2008). The Penguin Book of Card Games. Penguin UK. Online
  27. Parlett, D. (1990). The Oxford Guide to Card Games. New York: Oxford University Press.
  28. Carter, R. (2014). The History of Roulette. Liverpool Echo. Online
  29. Aris, S. (2021) Bernoulli’s Golden Theorem in Retrospect: Error Probabilities and Trustworthy Evidence. Synthese.
  30. Eglin, J. (2023). The Gambling Century: Commercial Gaming in Britain from Restoration to Regency. Oxford University Press. Online
  31. Hoyle, E. (1742) A Short Treatise on the Game of Whist. Bath ; London : For W. Webster. Online
  32. Harris, B. (2022). Gambling in Britain in the Long Eighteenth Century. Cambridge University Press. Online

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