Title Probability and Statistics Randomness Meteorology Dice Odds Probability Theory 1.5 MB 241
```                            Cover
Front Matter
Acknowledgments
Introduction
Part One: What is Probability?
1. The Idea of Randomness
Randomness before the Theory of Probability
Early Difficulties in Developing a Theory of Randomness
Randomness and Religion Today in Burkina Faso
2. The Nature of Chance
Cardano's Mistake
Cardano on Luck and Math
Galileo Galilei
Peirre de Fermat and Blaise Pascal
The Division of Stakes, an Alternative Interpretation
Christian Huygens
Jacob Bernoulli
Abraham de Moivre
De Moivre on Math and Luck
The Bell Curve
3. Surprising Insights into Probability and Its Uses
Thomas Bayes and Inverse Probability
Buffon and the Needle Problem
Daniel Bernoulli and Smallpox
Jean le Rond d'Alembert and the Evaluation of Risk
Leonhard Euler and Lotteries
4. Randomness in a Deterministic Universe
Simeon-Denis Poisson
The Poisson Distribution
5. Random Processes
James Clerk Maxwell
Brownian Motion Revisited
Markov Processes
A Markov Chain
6. Probability as a Mathematical Discipline
Theory and Practice
7. Three Applications of the Theory of Probability
Nuclear Reactor Safety
Markov Chains and Information Theory
Smallpox in Modern Historical Times
Part Two: Statistics
Introduction: The Age of Information
8. The Beginnings of Statistics
The Beginnings of Statistics
Edmund Halley
Breslau Table for Number and Infinity
Insurance
9. Data Analysis and the Problem of Precision
The Misuse of Statistics
10. The Birth of Modern Statistics
Karl Pearson
R.A. Fisher
11. The Theory of Sampling
The Problem
Walter Shewhart and Statistical Quality Control
William Edwards Deming
12. Three Applications of Statistics
The Birth of Epidemiology
The U.S. Census
Political Polling
Chronology
Glossary
Index
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##### Document Text Contents
Page 2

probability and
statistics

the science of uncertainty

Page 120

Shannon’s discovery of this definition of information allowed
him to show that information as he defined it obeys certain laws that
are in some ways analogous to those laws that describe the rate of
change of other physical quantities such as mass, momentum, and
energy. By employing the theory of probability, and especially the
theory of Markov chains, he was able to show that information can
be transmitted with extremely high accuracy even when the chan-
nel is noisy, provided that the information is correctly encoded at
the transmitter. This was a surprising result since before
Shannon’s work, it was generally assumed that on a noisy channel
parts of the transmitted message would inevitably be lost.
Shannon’s discovery led to the search for optimal error-correcting
codes, codes that were as fast as theoretically possible and that still
preserved the message in the presence of noise. Error-correcting
codes are now routinely used throughout our society. They make
it possible, for example, for the Voyager space probes, now located
at the farthest reaches of our solar system, to continue to commu-
nicate with Earth successfully by using 23-watt radios. More gen-
erally, Shannon’s discoveries are the foundation of all work in
digital communication, because they made it possible to develop a
successful mathematical model for the transmission and storage of
information.

Smallpox in Modern Historical Times
Smallpox has long been a public health scourge. It had been a
major source of mortality in the Eastern Hemisphere for thousands

Three Applications of the Theory of Probability 105

Information
source

Encoder
and

transmitter
Channel

and

decoder

Noise

A model depicting how information is transmitted

Page 121

of years. It spread throughout North, Central, and South America
with the arrival of European colonists, explorers, and conquerors.
The disease is rapidly transmitted between individuals, and there
has never been an effective treatment for those unfortunate enough
to become infected. Generally, about one-third of all those who
were infected died, although among Native Americans, in particu-
lar, the fatality rate was much higher. Throughout history a great
deal of thought has been given to controlling and eliminating
smallpox.

The technique of variolation, the dangerous but often effective
technique of conferring resistance to smallpox, was of profound
importance. (See the section on Daniel Bernoulli and Jean
d’Alembert earlier in this volume for background on variolation.)
One of the peculiar aspects of variolation is that it depends on the
existence of individuals infected with smallpox so that the small-
pox “matter” from the infected individual can be used to induce

106 PROBABILITY AND STATISTICS

Smallpox vaccination program, 1946, Jewell Ridge, Virginia (Courtesy
National Archives, College Park, Maryland)

Page 240

Shannon, Claude, mathe-
matical contribution of
92, 103–105

Shewhart, Walter
associates of 160
influence of 164
life of 153–154
publications 160
scientific contribution
of 153, 155–159
power and limitations
of 159–160

Shewhart charts 157,
157–158
acceptance of 163–164
power and limitations
of 159–160

sensitivity of 158
significance, testing 145
slavery, U.S. Census and
173–174

smallpox
cause of 54
elimination of

globally 108
in United States
107

history of
in premodern era
54–55, 105–106

in modern era
106–108

public health policy on,
probability analysis
and 54–57

16th-century work on
53

symptoms and mortality
rate in 55, 55, 106

vaccination programs
cost of 108
evaluation of risks
associated with
107–108

success of 106, 107,
108

vaccine for, discovery of
55, 68, 107

variolation for
definition of 55
need for infected per-
sons in 106–107

risks of 56, 60

evaluating with
probability analy-
sis 56–57, 68

issues of interpreta-
tion in 58–60

theory behind
55–56

vs. vaccination 68
as weapon, analysis of
threat 108–109

Smoluchowski, Marian
81–83

Snow, John 169–171
social sciences

applications of statistics
in 136–137

sampling of opinion.
See polling

Some Theory of Sampling
(Deming) 161, 163, 181

space shuttle program,
catastrophic failure
analysis in 95

special cause variation, in
manufacturing
characteristics of
155–156

definition of 155
distinguishing from
other variation types
157, 157–159

stability, of manufacturing
process
controlling 157,
157–159

difficulty of maintaining
155–156, 158

standardization, in manu-
facturing, development
of 151–152

Statistical Methods for
Research Workers (Fisher)
143

statistics
applications of. See
applications of statis-
tics

as applied discipline
114

characteristics of scien-
tists contributing to
137

data for. See data

founding of discipline
117–129

history of 44
influence of 182
mathematics in 114,
128–129

misuse of 134–135
modern, development
of 136–149

precision in
130

method of least
squares, develop-
ment of 131–133

prestatistical age, data in
115–117

purpose of 114
in refinement of experi-
mental design
145–147

sampling. See sampling;
sampling theory

significance, testing of
145

stochastic processes. See
random processes

stock market, trend analy-
sis applications 93

“A System of Profound
Knowledge” (Deming)
164

T
Taylor, Frederick Winslow
152

telephone networks, prob-
ability models in 74

terrorism
bioterrorism attack,
risk analysis for
108–109

September 11, 2001,
terrorist attacks,
impact of 108

theorems, basis of 91
“Theoria Motus
Corporum in Sectionibus
Conicus Solem
Ambientium” (Gauss)
132–133

Index 225

Page 241

Théorie analytique des prob-
abilités (Laplace) 64, 67

Theory of Errors (Airy)
142

Theory of Probability
(Jeffreys) 99

thila 12
Games” (Galileo) 21–24,
22

tides
atmospheric, analysis of
69–71

neap 70
ocean, causes of 70

traffic networks, probabil-
ity models in 74

transition probabilities 86
transmission of informa-
tion. See information
theory

A Treatise of Annuities on
Lives (de Moivre) 44

Treatise on the Small Pox
and Measles (ar-Razi) 54

The Trinity: A Nineteenth
Century Passion Play
(Pearson) 137

Trojan Nuclear Power
Plant 102

Truman, Harry 179,
179–180

Tyler, John 11

U
uncertainty, as inherent in
randomness xiii–xiv, 98

Uniformity System of
manufacturing 151

United States
Articles of
Confederation
171–172

cholera in 168
Constitution of

census provisions in
171, 172–173

representation in 172
universe

in polling, importance
of defining 180–181

in sampling, importance
of defining 161–162

unknowable aspects of
nature 66, 75

unquantifiable products,
quality control for 159

Ur, Royal Game of 7, 7–8
U.S. v. Holmes (1842) 9–11

V
vaccination

for smallpox
cost of 108
evaluation of risks
associated with
107–108

success of 106, 107,
108

vs. variolation 68
vaccine, for smallpox, dis-
covery of 55, 68, 107

variance, definition of 68,
134

variation
in manufacturing
process
control of 157,
157–159

types of 155–156
significance of, testing
145

variolation
definition of 55
need for infected per-
sons in 106–107

risks of 56, 60
evaluating with prob-
ability analysis
56–57, 68

issues of interpreta-
tion in 58–60

theory behind 55–56
vs. vaccination 68

velocity distribution 79
Venn, John 97–98
Venn diagrams 90, 97
Vibrio cholerae 165
Viète, François 150
Vitellius (emperor of
Rome) 8

volatile electorate, polling
and 180

volume, measurement
theory and 88–89

voters, identification of,
in polling 180–181

Voyager space probe 105

W
Wallis, John 150
weather forecasting

Halley’s contributions
to 122

probability in xii–xiii,
91

Weldon, Walter Frank
Raphael 140–141

Western Electric
Company 155, 158,
160

170–171

Whitney, Eli 151
WHO. See World Health
Organization

William Brown (ship) 10
William the Conqueror
115–117

World Health
Organization (WHO),
smallpox and 108