In the 17th century, Pascal’s strategy for solving problems of chance became the standard one. It was, for example, used by the Dutch mathematician Christiaan Huygens in his short treatise on games of chance, published in 1657. Huygens refused to define equality of chances as a fundamental presumption of a fair game but derived it instead from what he saw as a more basic notion of an equal exchange. Most questions of probability in the 17th century were solved, as Pascal solved his, by redefining the problem in terms of a series of games in which all players have equal expectations. The new theory of chances was not, in fact, simply about gambling but also about the legal notion of a fair contract. A fair contract implied equality of expectations, which served as the fundamental notion in these calculations. Measures of chance or probability were derived secondarily from these expectations.

Probability was tied up with questions of law and exchange in one other crucial respect. Chance and risk, in aleatory contracts, provided a justification for lending at interest, and hence a way of avoiding Christian prohibitions against usury. Lenders, the argument went, were like investors; having shared the risk, they deserved also to share in the gain. For this reason, ideas of chance had already been incorporated in a loose, largely nonmathematical way into theories of banking and marine insurance. From about 1670, initially in the Netherlands, probability began to be used to determine the proper rates at which to sell annuities. Jan de Wit, leader of the Netherlands from 1653 to 1672, corresponded in the 1660s with Huygens, and eventually he published a small treatise on the subject of annuities in 1671.

Annuities in early modern Europe were often issued by states to raise money, especially in times of war. They were generally sold according to a simple formula such as “seven years purchase,” meaning that the annual payment to the annuitant, promised until the time of his or her death, would be one-seventh of the principal. This formula took no account of age at the time the annuity was purchased. Wit lacked data on mortality rates at different ages, but he understood that the proper charge for an annuity depended on the number of years that the purchaser could be expected to live and on the presumed rate of interest. Despite his efforts and those of other mathematicians, it remained rare even in the 18th century for rulers to pay much heed to such quantitative considerations. Life insurance, too, was connected only loosely to probability calculations and mortality records, though statistical data on death became increasingly available in the course of the 18th century. The first insurance society to price its policies on the basis of probability calculations was the Equitable, founded in London in 1762.

Probability as the logic of uncertainty

The English clergyman Joseph Butler, in his very influential Analogy of Religion (1736), called probability “the very guide of life.” The phrase, however, did not refer to mathematical calculation but merely to the judgments made where rational demonstration is impossible. The word probability was used in relation to the mathematics of chance in 1662 in the Logic of Port-Royal, written by Pascal’s fellow Jansenists, Antoine Arnauld and Pierre Nicole. But from medieval times to the 18th century and even into the 19th, a probable belief was most often merely one that seemed plausible, came on good authority, or was worthy of approval. Probability, in this sense, was emphasized in England and France from the late 17th century as an answer to skepticism. Man may not be able to attain perfect knowledge but can know enough to make decisions about the problems of daily life. The new experimental natural philosophy of the later 17th century was associated with this more modest ambition, one that did not insist on logical proof.

Almost from the beginning, however, the new mathematics of chance was invoked to suggest that decisions could after all be made more rigorous. Pascal invoked it in the most famous chapter of his Pensées, “Of the Necessity of the Wager,” in relation to the most important decision of all, whether to accept the Christian faith. One cannot know of God’s existence with absolute certainty; there is no alternative but to bet (“il faut parier”). Perhaps, he supposed, the unbeliever can be persuaded by consideration of self-interest. If there is a God (Pascal assumed he must be the Christian God), then to believe in him offers the prospect of an infinite reward for infinite time. However small the probability, provided only that it be finite, the mathematical expectation of this wager is infinite. For so great a benefit, one sacrifices rather little, perhaps a few paltry pleasures during one’s brief life on Earth. It seemed plain which was the more reasonable choice.

The link between the doctrine of chance and religion remained an important one through much of the 18th century, especially in Britain. Another argument for belief in God relied on a probabilistic natural theology. The classic instance is a paper read by John Arbuthnot to the Royal Society of London in 1710 and published in its Philosophical Transactions in 1712. Arbuthnot presented there a table of christenings in London from 1629 to 1710. He observed that in every year there was a slight excess of male over female births. The proportion, approximately 14 boys for every 13 girls, was perfectly calculated, given the greater dangers to which young men are exposed in their search for food, to bring the sexes to an equality of numbers at the age of marriage. Could this excellent result have been produced by chance alone? Arbuthnot thought not, and he deployed a probability calculation to demonstrate the point. The probability that male births would by accident exceed female ones in 82 consecutive years is (0.5)⁸². Considering further that this excess is found all over the world, he said, and within fixed limits of variation, the chance becomes almost infinitely small. This argument for the overwhelming probability of Divine Providence was repeated by many—and refined by a few. The Dutch natural philosopher Willem ’sGravesande incorporated the limits of variation of these birth ratios into his mathematics and so attained a still more decisive vindication of Providence over chance. Nicolas Bernoulli, from the famous Swiss mathematical family, gave a more skeptical view. If the underlying probability of a male birth was assumed to be 0.5169 rather than 0.5, the data were quite in accord with probability theory. That is, no Providential direction was required.

Apart from natural theology, probability came to be seen during the 18th-century Enlightenment as a mathematical version of sound reasoning. In 1677 the German mathematician Gottfried Wilhelm Leibniz imagined a utopian world in which disagreements would be met by this challenge: “Let us calculate, Sir.” The French mathematician Pierre-Simon de Laplace, in the early 19th century, called probability “good sense reduced to calculation.” This ambition, bold enough, was not quite so scientific as it may first appear. For there were some cases where a straightforward application of probability mathematics led to results that seemed to defy rationality. One example, proposed by Nicolas Bernoulli and made famous as the St. Petersburg paradox, involved a bet with an exponentially increasing payoff. A fair coin is to be tossed until the first time it comes up heads. If it comes up heads on the first toss, the payment is 2 ducats; if the first time it comes up heads is on the second toss, 4 ducats; and if on the nth toss, 2ⁿ ducats. The mathematical expectation of this game is infinite, but no sensible person would pay a very large sum for the privilege of receiving the payoff from it. The disaccord between calculation and reasonableness created a problem, addressed by generations of mathematicians. Prominent among them was Nicolas’s cousin Daniel Bernoulli, whose solution depended on the idea that a ducat added to the wealth of a rich man benefits him much less than it does a poor man (a concept now known as decreasing marginal utility; see utility and value: Theories of utility).

Probability arguments figured also in more practical discussions, such as debates during the 1750s and ’60s about the rationality of smallpox inoculation. Smallpox was at this time widespread and deadly, infecting most and carrying off perhaps one in seven Europeans. Inoculation in these days involved the actual transmission of smallpox, not the cowpox vaccines developed in the 1790s by the English surgeon Edward Jenner, and was itself moderately risky. Was it rational to accept a small probability of an almost immediate death to reduce greatly a large probability of death by smallpox in the indefinite future? Calculations of mathematical expectation, as by Daniel Bernoulli, led unambiguously to a favourable answer. But some disagreed, most famously the eminent mathematician and perpetual thorn in the flesh of probability theorists, the French mathematician Jean Le Rond d’Alembert. One might, he argued, reasonably prefer a greater assurance of surviving in the near term to improved prospects late in life.