Probability gives a measure of likelihood or frequency of occurrence. If we throw a pair of dice 1,000
times, then we would expect, on average, close to onesixth of the rolls (around 167 of them) to have a sum
of 7; close to 1/12th of them (around 83 rolls) to have
sum 10; and close to 1/36th (around 28 of them) to
have a sum of 2. This principle is clarified in the LAW
OF LARGE NUMBERS.
Computing Probabilities
Two models are often used to help compute probabilities in moderately complicated situations. They can
also be used to illustrate two rules of computation.
1. Probability Trees and the Addition Rule
A probability tree is a diagram displaying all the outcomes of a sequence of actions. It is assumed that the
probability 415
SON (1781–1840) who, among other things, mathematically
proved the law of large numbers. The most important publication in this era on the theory of probability was Laplace’s
1812 text Théorie analytique des probabilities (Analytical
theory of probability.). In it, Laplace collected and extended
everything known on the subject at that time. Russian mathematicians PAFNUTY CHEBYSHEV (1821–94), Andrei Markov
(1856–1922), and Alexandr Lyapunov (1857–1918) further
developed the mathematical underpinnings of the subject in
the late 19th century.
Basic statistical thought can be deemed as having
developed considerably earlier. The ancient Egyptians
compiled DATA concerning population and wealth as early
as 3050 B.C.E., developing simple techniques to collate and
record the numerical information gathered. The ancient
Chinese undertook similar studies around 2300 B.C.E. A
census was taken in 594 B.C.E. by the Greeks for the purpose of levying taxes, and Athens undertook a population
census in 309 B.C.E. The Romans also kept census records,
as well as records of births and deaths, and gathered significant quantities of numerical information from geographic surveys taken across the entire empire. Very few
statistical records were kept during the period of the Middle Ages, however.
In 1662 John Graunt analyzed birth and death records
and produced the first LIFE TABLE. The purpose of the table
was to make general observations and predictions about
life expectancy for classes of members of a particular population. This work represented a significant step toward
analyzing data for the purposes of INFERENCE.
In 1790 the United States took its first decennial census, heralding the return of census taking. Several European nations followed suit soon afterward. Belgian
scholar LAMBERT ADOLPHE QUÉTELET (1796–1874) analyzed
the nation’s records and made important observations
about the influence of age, gender, occupation, and economic condition on mortality. In 1835 he attempted to apply
probabilistic methods to the study of human characteristics, both physical and behavioral. He used them to give
what he hoped was a complete description of the “average man.” Although Quételet’s work was generally highly
respected, his attempt to apply it to the field of behavioral
science was met with criticism. In the 1860s, English
scholar FRANCIS GALTON (1822–1911) attempted to apply
statistics methods to the study of human heredity. His
work was influential and helped define statistics as a
mathematics discipline in its own right.
At the turn of the 20th century, the corporate world
began to recognize the relevance and usefulness of statistics, especially in issues of quality control, economics,
insurance, and telecommunications. Many large companies
began hiring statisticians.
While working for an English brewing company, industrial scientist WILLIAM SEALY GOSSET (1876–1937) developed
the STUDENT’S T-TEST, allowing for the ability to derive reliable information from small samples. (Company policy forbade its employees to publish. Gosset did so in any case,
writing under the pseudonym “Student.”) English mathematician KARL PEARSON (1857–1936) developed the CHISQUARED TEST and is considered the founder of modern
hypothesis testing.
RONALD AYLMER FISHER (1890–1962) is considered the
most important statistician of the 20th century. His 1925
text Statistical Methods for Research Workers transformed statistics into a powerful scientific tool. He clarified many of the mathematical principles on which the
discipline is based. Fisher also developed methods of multivariate analysis to properly analyze problems involving
more than one variable.
In 1926, pure and applied mathematician JOHN VON
NEUMANN (1903–57) founded GAME THEORY—a mathematical
framework for analyzing games of chance, such as poker,
that involve strategy and choice on the parts of the players.
Von Neumann recognized the applications of the theory to
economics and social sciences. The work of Nobel Laureate JOHN FORBES NASH, JR., (born 1928) took its applications
to economics to a profound level.
See also STATISTICS: DESCRIPTIVE; STATISTICS: INFERENTIAL.
times, then we would expect, on average, close to onesixth of the rolls (around 167 of them) to have a sum
of 7; close to 1/12th of them (around 83 rolls) to have
sum 10; and close to 1/36th (around 28 of them) to
have a sum of 2. This principle is clarified in the LAW
OF LARGE NUMBERS.
Computing Probabilities
Two models are often used to help compute probabilities in moderately complicated situations. They can
also be used to illustrate two rules of computation.
1. Probability Trees and the Addition Rule
A probability tree is a diagram displaying all the outcomes of a sequence of actions. It is assumed that the
probability 415
SON (1781–1840) who, among other things, mathematically
proved the law of large numbers. The most important publication in this era on the theory of probability was Laplace’s
1812 text Théorie analytique des probabilities (Analytical
theory of probability.). In it, Laplace collected and extended
everything known on the subject at that time. Russian mathematicians PAFNUTY CHEBYSHEV (1821–94), Andrei Markov
(1856–1922), and Alexandr Lyapunov (1857–1918) further
developed the mathematical underpinnings of the subject in
the late 19th century.
Basic statistical thought can be deemed as having
developed considerably earlier. The ancient Egyptians
compiled DATA concerning population and wealth as early
as 3050 B.C.E., developing simple techniques to collate and
record the numerical information gathered. The ancient
Chinese undertook similar studies around 2300 B.C.E. A
census was taken in 594 B.C.E. by the Greeks for the purpose of levying taxes, and Athens undertook a population
census in 309 B.C.E. The Romans also kept census records,
as well as records of births and deaths, and gathered significant quantities of numerical information from geographic surveys taken across the entire empire. Very few
statistical records were kept during the period of the Middle Ages, however.
In 1662 John Graunt analyzed birth and death records
and produced the first LIFE TABLE. The purpose of the table
was to make general observations and predictions about
life expectancy for classes of members of a particular population. This work represented a significant step toward
analyzing data for the purposes of INFERENCE.
In 1790 the United States took its first decennial census, heralding the return of census taking. Several European nations followed suit soon afterward. Belgian
scholar LAMBERT ADOLPHE QUÉTELET (1796–1874) analyzed
the nation’s records and made important observations
about the influence of age, gender, occupation, and economic condition on mortality. In 1835 he attempted to apply
probabilistic methods to the study of human characteristics, both physical and behavioral. He used them to give
what he hoped was a complete description of the “average man.” Although Quételet’s work was generally highly
respected, his attempt to apply it to the field of behavioral
science was met with criticism. In the 1860s, English
scholar FRANCIS GALTON (1822–1911) attempted to apply
statistics methods to the study of human heredity. His
work was influential and helped define statistics as a
mathematics discipline in its own right.
At the turn of the 20th century, the corporate world
began to recognize the relevance and usefulness of statistics, especially in issues of quality control, economics,
insurance, and telecommunications. Many large companies
began hiring statisticians.
While working for an English brewing company, industrial scientist WILLIAM SEALY GOSSET (1876–1937) developed
the STUDENT’S T-TEST, allowing for the ability to derive reliable information from small samples. (Company policy forbade its employees to publish. Gosset did so in any case,
writing under the pseudonym “Student.”) English mathematician KARL PEARSON (1857–1936) developed the CHISQUARED TEST and is considered the founder of modern
hypothesis testing.
RONALD AYLMER FISHER (1890–1962) is considered the
most important statistician of the 20th century. His 1925
text Statistical Methods for Research Workers transformed statistics into a powerful scientific tool. He clarified many of the mathematical principles on which the
discipline is based. Fisher also developed methods of multivariate analysis to properly analyze problems involving
more than one variable.
In 1926, pure and applied mathematician JOHN VON
NEUMANN (1903–57) founded GAME THEORY—a mathematical
framework for analyzing games of chance, such as poker,
that involve strategy and choice on the parts of the players.
Von Neumann recognized the applications of the theory to
economics and social sciences. The work of Nobel Laureate JOHN FORBES NASH, JR., (born 1928) took its applications
to economics to a profound level.
See also STATISTICS: DESCRIPTIVE; STATISTICS: INFERENTIAL.