Chapter 5: Case Study 1 — Pascal's Wager and the Birth of Probability Theory

The Dice Problem That Changed How Humans Think About Chance

Overview

In the summer of 1654, two French intellectuals exchanged a series of letters about gambling. The letters were technical, courteous, and intellectually explosive. Within them, almost without intending to, Pascal and Fermat solved a problem that had stumped mathematicians for a century — and in solving it, established the conceptual foundations of probability theory, the discipline that would eventually make insurance, epidemiology, finance, and the science of luck itself possible.

This case study traces that correspondence, examines the mathematical problems they solved, considers the philosophical implications for how humanity thinks about chance, and connects the story to a separate but related contribution of Pascal's: the famous "wager" that used probability reasoning to argue for religious belief. In the intersection of gambling mathematics and theology, Pascal occupies a unique and revealing position in the history of luck.

Part I: The Chevalier de Méré's Problem

The Players

Antoine Gombaud, the Chevalier de Méré, was one of the more interesting figures in 17th-century French intellectual life — a nobleman, essayist, and enthusiastic gambler who took gambling seriously enough to think mathematically about it, without being quite good enough at mathematics to resolve his own questions. He was also a friend and admirer of Blaise Pascal, and in 1654 he brought Pascal two gambling problems that had been troubling him.

The Chevalier's problems were practically motivated — he lost money on one of his favorite bets and could not understand why the mathematics seemed to contradict his intuition. But they were also intellectually important, because they pointed to gaps in mathematical understanding that had not yet been filled.

Blaise Pascal (1623–1662) was, at the time of the correspondence, in his early thirties and already one of the most accomplished mathematicians and physicists in France. He had invented one of the first mechanical calculators at age eighteen. He had done foundational work in projective geometry and fluid mechanics. He had conducted the famous "vacuum" experiments that bear his name. He was also deeply preoccupied, in this period, with questions of religion and the relationship between reason and faith.

Pierre de Fermat (1601–1665) was a jurist by profession — he spent his working life as a parliamentary lawyer in Toulouse — and a mathematician by passionate avocation. Despite never holding a university position and publishing almost none of his mathematical work during his lifetime, Fermat is one of the most significant mathematicians in history, known for Fermat's Last Theorem, Fermat's Little Theorem, the method of infinite descent, and contributions to analytic geometry that parallel and predate some of Descartes's work.

These two men, connected by intellectual admiration and both responding to the practical questions of a gambling nobleman, were about to do something that would change the history of thought.

Part II: The Problem of Points

The Question

The "problem of points" (French: problème des points) is this:

Two players, Player A and Player B, are playing a game of pure chance — for our purposes, a coin flip game. Each correct call wins a point. The first to reach some target number of points (say, six) wins the pot. They have wagered equal amounts.

Now suppose the game is interrupted before anyone has won. At the moment of interruption, Player A has five points and Player B has three. Neither has reached six. How should they divide the pot fairly?

This sounds simple. It is not.

The naive answer — divide the pot in proportion to current scores, 5:3 — is wrong. Player A deserved more than a 5/8 share of the pot, because Player A only needed one more point to win while Player B needed three. The ratio of current scores does not capture the relevant information.

The correct answer requires reasoning about possible futures — about what would happen if the game continued from the current position. And reasoning about possible futures under uncertainty is exactly what probability theory provides.

The Confusion Before Pascal and Fermat

The problem had been discussed before. Luca Pacioli, an Italian mathematician, addressed it in 1494 — and got it wrong, using the naive proportional division. Niccolo Tartaglia, in 1556, pointed out that Pacioli's solution was incorrect but could not provide a satisfactory alternative. Girolamo Cardano grappled with it. Galileo touched on related problems. But no one had produced a general, correct solution.

The difficulty was conceptual, not merely computational: the problem required thinking systematically about hypothetical future outcomes — about what might happen, not what had happened. Mathematical tradition, inheriting from Aristotle, was comfortable with what was actual. Reasoning about what might have happened in a counterfactual future was philosophically novel territory.

Pascal and Fermat's Solutions

The two mathematicians approached the problem differently — and both arrived at the same answer, confirming that they had found the true solution.

Fermat's approach was combinatorial. He enumerated all the possible futures from the current game state. From the position of 5 to 3 (needing 1 more point versus 3 more points), there are at most three more games needed. Fermat listed all the possible outcomes of those three games:

  • AAA (A wins all three remaining games) — A wins
  • AAB (A wins first two, B wins last) — A wins (already reached 6 before third game)
  • ABA — A wins
  • ABB — A wins (fifth needed win already came)
  • BAA — A wins
  • BAB — A wins
  • BBA — A wins
  • BBB — B wins (the only scenario where B reaches 6 first)

Of the eight equally likely possible scenarios, A wins in 7 and B wins in 1. Therefore, A should receive 7/8 of the pot.

This was Fermat's method: enumerate all possible outcomes, count how many favor each player, and divide accordingly.

Pascal's approach was recursive. He developed a method of reasoning about the value of each future game from the current state backward, determining what each player "deserves" at each moment. His method, expressed through what we now call the "Pascal's triangle" and its relationship to combinations, produced a more general formula that could handle any game state without needing to enumerate every future.

Both approaches required the same foundational insight: that the fair division of stakes in an interrupted game is determined by the probability distribution of future outcomes, not by any feature of past outcomes. This seems obvious in retrospect. It was not obvious at all in 1654.

Part III: The Mathematics They Built

What the Correspondence Established

The Pascal-Fermat correspondence on the problem of points established, in embryonic form, several foundational concepts:

Expected value: The idea that the fair price of a gamble is the probability-weighted average of its possible outcomes. If you have a 7/8 chance of winning 100 and a 1/8 chance of winning 0, the expected value is (7/8 × 100) + (1/8 × 0) = 87.5. You should be willing to pay up to 87.5 for the right to take this gamble.

Combinatorial probability: The systematic method of counting possible outcomes to determine probabilities, using what we now call combinatorics (the mathematics of counting).

The principle of equivalent situations: If two game situations have the same probability distribution over future outcomes, they are equivalent from the perspective of expected value, regardless of how they were reached.

Christiaan Huygens Extends the Work

Huygens, the Dutch physicist and mathematician, learned of the Pascal-Fermat work through correspondence and published the first formal text on probability, De Ratiociniis in Ludo Aleae ("On Reasoning in Games of Chance"), in 1657. His text introduced the concept of expected value explicitly and provided a systematic treatment of several gambling problems. This text was the standard reference on probability for fifty years.

Jacob Bernoulli Completes the Foundation

The law of large numbers — perhaps the most important foundational result in probability theory — was proved by Jacob Bernoulli (uncle of the more famous Daniel Bernoulli) in his posthumously published Ars Conjectandi (1713). The law states, roughly, that as the number of trials of a random event increases, the observed frequency of each outcome converges to its probability. In other words: in the long run, the fair coin does come up heads approximately half the time. This result, which seems intuitive, required careful mathematical proof — and its proof was one of the most important achievements in the history of mathematics.

Part IV: Pascal's Wager — Probability Reasoning Applied to Faith

The Context

By the time Pascal wrote the Pensées — his fragmentary notes toward a defense of Christianity, published posthumously in 1670 — he was deeply preoccupied with both probability mathematics and religious faith. In the famous section known as the "Wager," he applied, probably for the first time in history, expected value reasoning to a theological question.

The Argument

Pascal's Wager goes, in simplified form, like this:

Consider two possibilities: either God exists or God does not exist. You must bet — you cannot abstain from the choice, because choosing not to bet on God is itself a choice (a bet against God's existence, or at minimum against the importance of religious practice).

Now consider the payoffs:

  • If God exists and you believe: infinite gain (eternal life)
  • If God exists and you do not believe: infinite loss (eternal damnation, or at minimum exclusion from eternal life)
  • If God does not exist and you believe: finite loss (some time and effort spent on religious practice)
  • If God does not exist and you do not believe: finite gain (time and effort saved)

Even if the probability that God exists is very small — even infinitesimally small — the expected value of believing is infinite (because infinite gain multiplied by any positive probability is infinite), while the expected value of not believing is negative infinite or very small positive. Therefore, the rational bet is to believe.

The Reception and the Problems

Pascal's Wager was immediately controversial and has been debated by philosophers for three and a half centuries. The main objections include:

The many-gods problem: Pascal assumes that the relevant choice is between the specific Christian God and no God. But there are many possible deities and theological systems, some of which punish belief in the wrong deity more severely than unbelief. If you cannot determine which god to wager on, the Wager's logic collapses.

The sincerity problem: Can you genuinely choose to believe something? Pascal addressed this by suggesting that you could begin with the practices of belief (attending church, praying, living as a believer) and that genuine belief would follow. But critics argue that belief produced by calculation is not the kind of belief that a God who values genuine faith would reward.

The probability problem: Pascal treats the probability that God exists as non-zero without providing a method for estimating it. If the probability is exactly zero, the expected value calculation collapses regardless of the infinite payoff. And "how do we estimate the probability of God's existence?" is not a question that probability theory alone can answer.

The utility problem: The notion of "infinite utility" is mathematically problematic. Infinite utilities produce paradoxes in expected value calculations — see the St. Petersburg Paradox, which Bernoulli also discussed in Ars Conjectandi.

What the Wager Tells Us About Probability's History

Despite its philosophical problems, Pascal's Wager is historically significant precisely because it represents the first attempt to apply probability reasoning — expected value calculation — to a domain entirely outside games of chance. Pascal was trying to show that the new mathematical tools developed for gambling were general tools for reasoning under uncertainty in any domain.

This is the seed of what would become decision theory, Bayesian epistemology, and cost-benefit analysis — all modern frameworks that use expected value reasoning to guide choices under uncertainty in domains from medicine to public policy.

The irony is rich: the mathematical tools invented by talking about dice games were first applied beyond games by a profoundly religious man trying to argue for faith. The journey from Fortuna to probability — from luck-as-goddess to luck-as-mathematics — passed through a theological wager that used gambling logic to argue for God.

Part V: The Philosophical Transformation

Before Pascal and Fermat: The Past as Explanation

Before the development of probability theory, explanations of chance events focused overwhelmingly on the past. Divination systems — oracle bones, entrails, astrology, omens — all attempted to read the causes of outcomes in existing signs. The fundamental orientation was backward-looking: what has happened (or what signs are present) that explains what will happen?

Even causal explanations that were not supernatural — like Aristotle's concept of tuche — were fundamentally about explaining why a particular outcome occurred, not about predicting a distribution of possible future outcomes.

After Pascal and Fermat: The Future as Probability Distribution

Probability theory introduced a new cognitive orientation: forward-looking, distributional, quantitative. Rather than asking "what caused this outcome?" it asked "what is the probability distribution of possible outcomes from here?" Rather than seeking a single explanation for a single event, it sought to characterize the full range of possible futures and their relative likelihoods.

This is a profound conceptual shift. It means that chance is not simply "things that don't have causes" — it is a structured relationship between the present and a distribution of possible futures. And structured relationships can be reasoned about mathematically.

The practical consequences were enormous. Life insurance companies could price policies based on mortality tables. Governments could plan for probable harvest failures. Ship owners could calculate whether insurance premiums were fairly priced. Engineers could design structures with quantified tolerances for stress. And eventually, scientists could design experiments that would distinguish signal from noise — the very foundation of modern empirical science.

Discussion Questions

  1. Pascal and Fermat were motivated by a practical gambling problem, not by an abstract interest in mathematical theory. Does it matter why a mathematical tool is developed, as long as the tool itself is useful? Can you think of other important intellectual tools developed for pragmatic or even disreputable purposes that turned out to be foundational?

  2. Pascal's Wager is often presented as an argument for religious belief, but it can also be read as an early application of decision theory. Setting aside the theological questions, what is valuable and what is problematic about using expected value reasoning to guide major life choices?

  3. The chapter describes the development of probability theory as "taking luck out of the realm of the divine and placing it in the realm of the mathematical." Was this a loss as well as a gain? What did probability theory replace that had genuine psychological or social value?

  4. Jacob Bernoulli's law of large numbers proved that observed frequencies converge to probabilities over many trials. How does this result change what we should believe about short-run "luck"? Does it mean short-run luck doesn't matter, or that it matters but is not meaningful?

See also: Chapter 6 — Probability Intuition, for a full treatment of how humans actually reason (and misreason) about probability, building on the mathematical foundations established in this case study.