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Statistics and Probability

❶Almost from the beginning, however, the new mathematics of chance was invoked to suggest that decisions could after all be made more rigorous.

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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 , 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 to , corresponded in the s with Huygens, and eventually he published a small treatise on the subject of annuities in Annuities in early modern Europe were often issued by states to raise money, especially in times of war. 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 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. 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. 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 and published in its Philosophical Transactions in Arbuthnot presented there a table of christenings in London from to 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. 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. 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.

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 the German mathematician Gottfried Wilhelm Leibniz imagined a utopian world in which disagreements would be met by this challenge: 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 n th toss, 2 n 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.

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 s 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. One might, he argued, reasonably prefer a greater assurance of surviving in the near term to improved prospects late in life. Jakob Bernoulli , uncle of Nicolas and Daniel, formulated and proved a law of large numbers to give formal structure to such reasoning. This was published in from a manuscript, the Ars conjectandi , left behind at his death in There he showed that the observed proportion of, say, tosses of heads or of male births will converge as the number of trials increases to the true probability p , supposing that it is uniform.

His theorem was designed to give assurance that when p is not known in advance, it can properly be inferred by someone with sufficient experience. He thought of disease and the weather as in some way like drawings from an urn. At bottom they are deterministic, but since one cannot know the causes in sufficient detail, one must be content to investigate the probabilities of events under specified conditions.

But Hartley named no names, and the first publication of the formula he promised occurred in in a posthumous paper of Thomas Bayes , communicated to the Royal Society by the British philosopher Richard Price. The result was perhaps more consequential in theory than in practice.

Laplace and his more politically engaged fellow mathematicians, most notably Marie-Jean-Antoine-Nicolas de Caritat, marquis de Condorcet , hoped to make probability into the foundation of the moral sciences. This took the form principally of judicial and electoral probabilities, addressing thereby some of the central concerns of the Enlightenment philosophers and critics. Justice and elections were, for the French mathematicians, formally similar.

In each, a crucial question was how to raise the probability that a jury or an electorate would decide correctly. One element involved testimonies, a classic topic of probability theory.

In the British mathematician John Craig used probability to vindicate the truth of scripture and, more idiosyncratically, to forecast the end of time, when, due to the gradual attrition of truth through successive testimonies, the Christian religion would become no longer probable.

The Scottish philosopher David Hume , more skeptically, argued in probabilistic but nonmathematical language beginning in that the testimonies supporting miracles were automatically suspect, deriving as they generally did from uneducated persons, lovers of the marvelous.

Miracles, moreover, being violations of laws of nature, had such a low a priori probability that even excellent testimony could not make them probable. Condorcet also wrote on the probability of miracles, or at least faits extraordinaires , to the end of subduing the irrational. But he took a more sustained interest in testimonies at trials, proposing to weigh the credibility of the statements of any particular witness by considering the proportion of times that he had told the truth in the past, and then use inverse probabilities to combine the testimonies of several witnesses.

Laplace and Condorcet applied probability also to judgments. In contrast to English juries, French juries voted whether to convict or acquit without formal deliberations. There would be no injustice, Condorcet argued, in exposing innocent defendants to a risk of conviction equal to risks they voluntarily assume without fear, such as crossing the English Channel from Dover to Calais.

Using this number and considering also the interest of the state in minimizing the number of guilty who go free, it was possible to calculate an optimal jury size and the majority required to convict.

But by this time the whole enterprise had come to seem gravely doubtful, in France and elsewhere. During the 19th century, statistics grew up as the empirical science of the state and gained preeminence as a form of social knowledge. Population and economic numbers had been collected, though often not in a systematic way, since ancient times and in many countries. In Europe the late 17th century was an important time also for quantitative studies of disease, population, and wealth.

In the English statistician John Graunt published a celebrated collection of numbers and observations pertaining to mortality in London, using records that had been collected to chart the advance and decline of the plague. Still, he did not leave such matters to nature and to God, but rather he offered abundant advice about how kings and princes could promote the growth of their populations.

He envisioned a rather spartan order of small farmers, paying modest rents and taxes, living without luxury, and practicing the Protestant faith.

Roman Catholicism was unacceptable on account of priestly celibacy. Lacking, as they did, complete counts of population, 18th-century practitioners of political arithmetic had to rely largely on conjectures and calculations. In France especially, mathematicians such as Laplace used probability to surmise the accuracy of population figures determined from samples. In the 19th century such methods of estimation fell into disuse, mainly because they were replaced by regular, systematic censuses.

The census of the United States, required by the U. Constitution and conducted every 10 years beginning in , was among the earliest. For the role of the U.

Sweden had begun earlier; most of the leading nations of Europe followed by the midth century. They were also eager to survey the populations of their colonial possessions, which indeed were among the very first places to be counted.

A variety of motives can be identified, ranging from the requirements of representative government to the need to raise armies. Some of this counting can scarcely be attributed to any purpose, and indeed the contemporary rage for numbers was by no means limited to counts of human populations. From the midth century and especially after the conclusion of the Napoleonic Wars in , the collection and publication of numbers proliferated in many domains, including experimental physics, land surveys, agriculture, and studies of the weather, tides, and terrestrial magnetism.

For perhaps the best statistical graph ever constructed, see the figure. Still, the management of human populations played a decisive role in the statistical enthusiasm of the early 19th century. Political instabilities associated with the French Revolution of and the economic changes of early industrialization made social science a great desideratum. A new field of moral statistics grew up to record and comprehend the problems of dirt, disease, crime, ignorance, and poverty.

Some of these investigations were conducted by public bureaus, but much was the work of civic-minded professionals, industrialists, and, especially after midcentury, women such as Florence Nightingale see the figure. One of the first serious statistical organizations arose in as section F of the new British Association for the Advancement of Science. The intellectual ties to natural science were uncertain at first, but there were some influential champions of statistics as a mathematical science.

The most effective was the Belgian mathematician Adolphe Quetelet , who argued untiringly that mathematical probability was essential for social statistics. Quetelet hoped to create from these materials a new science, which he called at first social mechanics and later social physics. He wrote often of the analogies linking this science to the most mathematical of the natural sciences, celestial mechanics. In practice, though, his methods were more like those of geodesy or meteorology, involving massive collections of data and the effort to detect patterns that might be identified as laws.

These, in fact, seemed to abound. He found them in almost every collection of social numbers, beginning with some publications of French criminal statistics from the mids. The numbers, he announced, were essentially constant from year to year, so steady that one could speak here of statistical laws. Even Quetelet had been startled at first by the discovery of these statistical laws.

Regularities of births and deaths belonged to the natural order and so were unsurprising, but here was constancy of moral and immoral acts, acts that would normally be attributed to human free will. Was there some mysterious fatalism that drove individuals, even against their will, to fulfill a budget of crimes?

Another site that advanced learners might try is The Birthday Problem. Finally, you can find several challenging problems at Probability in My Garden. Visit Games of Chance and proceed through the 8 modules. Then conduct the related experiments. Adapt or follow the procedures found at one or more of the following webQuest sites: Roller Coaster Statistics by A.

The Study of Probability Grade 6 by L. Theoretical Grades by L. Grades by T. Visit the Statistical Quote of the Day site.

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