Gregersen E. (editor) The Britannica Guide to Statistics and Probability
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CHAPTER 1
HIstoRY oF stAtIstICs
AnD PRoBABILItY
S
tatistics and probability are the branches of mathe-
matics concerned with the laws governing random
events, including the collection, analysis, interpretation,
and display of numerical data. Probability has its origin in
the study of gambling and insurance in the 17th century,
and it is now an indispensable tool of both social and natu-
ral sciences. Statistics may be said to have its origin in
census counts taken thousands of years ago. As a distinct
scientifi c discipline, however, it was developed in the early
19th century as the study of populations, economies, and
moral actions and later in that century as the mathemati-
cal tool for analyzing such numbers.
eaRly pRobabiliTy
It is astounding that for a subject that has altered how
humanity views nature and society, probability had its
beginnings in frivolous gambling. How much should you
bet on the turn of a card? An entirely new branch of math-
ematics developed from such questions.
Games of Chance
The modern mathematics of chance is usually dated to a
correspondence between the French mathematicians
Pierre de Fermat and Blaise Pascal in 1654. Their inspira-
tion came from a problem about games of chance,
proposed by a remarkably philosophical gambler, the che-
valier de Méré. De Méré inquired about the proper
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Blaise Pascal invented the syringe and created the hydraulic press, an instru-
ment based upon the principle that became known as Pascal’s law. Boyer/
Roger Viollet/Getty Images
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History of Statistics and Probability 7
division of the stakes when a game of chance is inter-
rupted. Suppose two players, A and B, are playing a
three-point game, each having wagered 32 pistoles, and are
interrupted after A has two points and B has one. How
much should each receive?
Fermat and Pascal proposed somewhat different solu-
tions, but they agreed about the numerical answer. Each
undertook to define a set of equal or symmetrical cases,
then to answer the problem by comparing the number for
A with that for B. Fermat, however, gave his answer in
terms of the chances, or probabilities. He reasoned that
two more games would suffice in any case to determine a
victory. There are four possible outcomes, each equally
likely in a fair game of chance. A might win twice, AA; or
first A then B might win; or B then A; or BB. Of these four
sequences, only the last would result in a victory for B.
Thus, the odds for A are 3:1, implying a distribution of 48
pistoles for A and 16 pistoles for B.
Pascal thought Fermat’s solution unwieldy, and he pro-
posed to solve the problem not in terms of chances but in
terms of the quantity now called “expectation.” Suppose B
had already won the next round. In that case, the positions
of A and B would be equal, each having won two games,
and each would be entitled to 32 pistoles. A should receive
his portion in any case. B’s 32, by contrast, depend on the
assumption that he had won the first round. This first
round can now be treated as a fair game for this stake of 32
pistoles, so that each player has an expectation of 16.
Hence A’s lot is 32 + 16, or 48, and B’s is just 16.
Games of chance such as this one provided model
problems for the theory of chances during its early period,
and indeed they remain staples of the textbooks. A post-
humous work of 1665 by Pascal on the “arithmetic triangle”
now linked to his name showed how to calculate numbers
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of combinations and how to group them to solve elemen-
tary gambling problems. Fermat and Pascal were not the
first to give mathematical solutions to problems such as
these. More than a century earlier, the Italian mathemati-
cian, physician, and gambler Girolamo Cardano calculated
odds for games of luck by counting up equally probable
cases. His little book, however, was not published until
1663, by which time the elements of the theory of chances
were already well known to mathematicians in Europe. It
will never be known what would have happened had
Cardano published in the 1520s. It cannot be assumed that
probability theory would have taken off in the 16th cen-
tury. When it began to flourish, it did so in the context of
the “new science” of the 17th-century scientific revolu-
tion, when the use of calculation to solve tricky problems
had gained a new credibility. Cardano, moreover, had no
great faith in his own calculations of gambling odds, since
he believed also in luck, particularly in his own. In the
Renaissance world of monstrosities, marvels, and simili-
tudes, chance—allied to fate—was not readily naturalized,
and sober calculation had its limits.
Risks, Expectations, and Fair Contracts
In the 17th century, Pascal’s strategy for solving problems
of chance became the standard. 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 funda-
mental 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 cen-
tury were solved, as Pascal solved his, by redefining the
problem in terms of a series of games in which all players
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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 prohibi-
tions 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 nonmathe-
matical 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 for-
mula 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 pur-
chaser could be expected to live and on the presumed rate
of interest. Despite his efforts and those of other mathe-
maticians, it remained rare even in the 18th century for
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rulers to pay much heed to such quantitative consider-
ations. 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 influen-
tial Analogy of Religion (1736), called probability “the very
guide of life.” The phrase did not refer to mathemati-
cal calculation, however, but merely to the judgments
made where rational demonstration is impossible. The
word probability was used in relation to the mathemat-
ics 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 author-
ity, 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 exper-
imental 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 mathe-
matics 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
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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 sup-
posed, the unbeliever can be persuaded by consideration
of self-interest. If there is a God (Pascal assumed he must
be the Christian God), to believe in him offers the pros-
pect of an infinite reward for infinite time. However small
the probability, provided only that it be finite, the mathe-
matical 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 cen-
tury, 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)
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.
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
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was repeated by many—and refined by a few. The Dutch
natural philosopher Willem’s Gravesande incorporated
the limits of variation of these birth ratios into his math-
ematics 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 math-
ematical 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 math-
ematician Pierre-Simon de Laplace, in the early 19th
century, called probability “good sense reduced to cal-
culation.” 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 math-
ematics 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 expo-
nentially 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, 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 reason-
ableness created a problem, addressed by generations of
mathematicians. Prominent among them was Nicolas’s
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Pierre Simon de Laplace demonstrated the usefulness of probability for inter-
preting scientific data. Hulton Archive/Getty Images