##### 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