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Publisher:

Dover Publications

Publication Date:

1982

Number of Pages:

392

Format:

Paperback

Price:

12.95

ISBN:

978-0486243429

Category:

Textbook

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by , on ]

Allen Stenger

08/29/2010

This 1964 book actually lives up to its back-cover blurb: “the best nontechnical introduction to probability ever written.” The intended audience is bright high-school students, lower-division undergraduates, and the proverbial “intelligent reader”.

“Nontechnical” is something of an exaggeration, as the book is full of numerical calculations, it uses a lot of high-school algebra, and there is no shortage of equations. The exposition focuses on examples with real data and shows how the real data match the probabilistic models. The author does an excellent job of picking which details to reveal and which to keep hidden, and he is always careful to point out where each concept is useful. It is a very concrete approach to the subject.

Despite the modest technical requirements, the book manages to cover a lot of ground, including expectation, Chebyshev’s inequality, binomial and Poisson distributions, the central limit theorem, and gambler’s ruin. The book deals strictly with discrete distributions, except for a brief treatment of the normal distribution as a limiting distribution. The book also gives an introduction to statistics, but instead of being the usual outline of descriptive statistics it focuses on coincidences and low-probability events, and how confidence intervals and hypothesis testing are useful.

This is not a textbook in a conventional sense, as the emphasis is on understanding the concepts and importance of probabilistic thinking rather than on acquiring techniques. There are a modest number of exercises, labeled “Try These Yourself”, which aim at solidifying understanding rather than presenting new challenges.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

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Foreword

I Thoughts about Thinking

The Reasoning Animal

Reasoning and Fun

The Kind of Questions We Have to Answer

What Kind of Reasoning Is Able to Furnish Useful Replies to Questions of This Sort

Thinking and Reasoning

Classical Logic

II The Birth of Lady Luck

III The Concept of Mathematical Probability

Don't Expect Too Much

Mathematical Theories and the Real World of Events

Mathematical Models

Can There Be Laws for Chance?

The Rolling of a Pair of Dice

The Number of Outcomes

Equally Probable Outcomes

Ways of Designing Models

The Definition of Mathematical Probability

A Recapitulation and a Look Ahead

Note on Terminology

Note on Other Books about Probability

IV The Counting of Cases

Preliminary

Compound Events

Permutations

Combinations

More Complicated Cases

V Some Basic Probability Rules

A Preliminary Warning

Independent Events and Mutually Exclusive Events

Converse Events

Fundamental Formulas for Total and for Compound Probability

VI Some Problems

Foreword

The First Problem of de Méré

The Problem of the Three Chests

A Few Classical Problems

The Birthday Problem

Montmort's Problem

Try These Yourself

Note about Decimal Expansions

VII Mathematical Expectation

How Can I Measure My Hopes?

Mathematical Expectation

The Jar with 100 Balls

The One-Armed Bandit

The Nicolas Bernoulli Problem

The St. Petersburg Paradox

Summary Remarks about Mathematical Expectation

Try These

Where Do We Eat?

VIII The Law of Averages

The Long Run

Heads or Tails

IX Variability and Chebychev's Theorem

Variability

Chebychev's Theorem

X Binomial Experiments

Binomial Experiments

"Why "Binomial"?"

Pascal's Arithmetic Triangle

Binomial Probability Theorem

Some Characteristics of Binomial Experiments

XI The Law of Large Numbers

Bernoulli's Theorem

Comments About the Classical Law of Large Numbers

Improved Central Limit Theorems

Note on Large Numbers

XII Distribution Functions and Probabilities

Probability Distributions

Normalized Charts

The Normal or Gaussian Distribution

What Is Normally Distributed?

The Quincunx

"Other Probability Distributions, The Poisson Distribution"

The Distribution of First Significant Digits

XIII "Rare Events, Coincidences, and Surprising Occurrences"

"Well, What Do You Think about That!"

Small Probabilities

Note on the Probability of Dealing Any Specified Hand of Thirteen Cards

Further Note on Rare Events

XIV Probability and Statistics

Statistics

Deduction and Induction

Sampling

What Sort of Answers Can Statistics Furnish?

The Variation of Random Samples

Questions (2) and (3): Statistical Inference

Question (4): Experimental Design

XV Probability and Gambling

The Game of Craps

The Ruin of the Player

"Roulette, Lotteries, Bingo, and the Like"

Gambling Systems

XVI Lady Luck Becomes a Lady

Preliminary

The Probability of an Event

Geometrical Probabilities

It Can't Be Chance!

The Surprising Stability of Statistical Results

The Subtlety of Probabilistic Reasoning

The Modern Reign of Probability

Lady Luck and the Future

Index

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