Probability gives a measure of likelihood or fre-
quency of occurrence. If we throw a pair of dice 1,000
times, then we would expect, on average, close to one-
sixth 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 probabili-
ties 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 out-
comes 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 publi-
cation 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 math-
ematicians P
AFNUTY
C
HEBYSHEV
(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 pur-
pose 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 sig-
nificant quantities of numerical information from geo-
graphic surveys taken across the entire empire. Very few
statistical records were kept during the period of the Mid-
dle 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 pop-
ulation. This work represented a significant step toward
analyzing data for the purposes of
INFERENCE
.
In 1790 the United States took its first decennial cen-
sus, heralding the return of census taking. Several Euro-
pean nations followed suit soon afterward. Belgian
scholar L
AMBERT
A
DOLPHE
Q
UÉTELET
(1796–1874) analyzed
the nation’s records and made important observations
about the influence of age, gender, occupation, and eco-
nomic condition on mortality. In 1835 he attempted to apply
probabilistic methods to the study of human characteris-
tics, both physical and behavioral. He used them to give
what he hoped was a complete description of the “aver-
age 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 F
RANCIS
G
ALTON
(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 statis-
tics, especially in issues of quality control, economics,
insurance, and telecommunications. Many large companies
began hiring statisticians.
While working for an English brewing company, indus-
trial scientist W
ILLIAM
S
EALY
G
OSSET
(1876–1937) developed
the
STUDENT
’
ST
-
TEST
, allowing for the ability to derive reli-
able information from small samples. (Company policy for-
bade its employees to publish. Gosset did so in any case,
writing under the pseudonym “Student.”) English mathe-
matician K
ARL
P
EARSON
(1857–1936) developed the
CHI
-
SQUARED TEST
and is considered the founder of modern
hypothesis testing.
R
ONALD
A
YLMER
F
ISHER
(1890–1962) is considered the
most important statistician of the 20th century. His 1925
text Statistical Methods for Research Workers trans-
formed statistics into a powerful scientific tool. He clari-
fied many of the mathematical principles on which the
discipline is based. Fisher also developed methods of mul-
tivariate analysis to properly analyze problems involving
more than one variable.
In 1926, pure and applied mathematician J
OHN VON
N
EUMANN
(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 Laure-
ate J
OHN
F
ORBES
N
ASH
, J
R
., (born 1928) took its applications
to economics to a profound level.
See also
STATISTICS
:
DESCRIPTIVE
;
STATISTICS
:
INFERENTIAL
.