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
Receiver
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
early concerns about
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
Stevenson, Adlai 178–179
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
“Thoughts about Dice-
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
Whitehead, Henry
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
World Trade Center
attacks (September 11,
2001), impact of 108
226 PROBABILITY AND STATISTICS
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
Receiver
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
early concerns about
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
Stevenson, Adlai 178–179
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
“Thoughts about Dice-
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
Whitehead, Henry
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
World Trade Center
attacks (September 11,
2001), impact of 108
226 PROBABILITY AND STATISTICS
