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.
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
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
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
More Complicated Cases
V Some Basic Probability Rules
A Preliminary Warning
Independent Events and Mutually Exclusive Events
Fundamental Formulas for Total and for Compound Probability
VI Some Problems
The First Problem of de Méré
The Problem of the Three Chests
A Few Classical Problems
The Birthday Problem
Try These Yourself
Note about Decimal Expansions
VII Mathematical Expectation
How Can I Measure My Hopes?
The Jar with 100 Balls
The One-Armed Bandit
The Nicolas Bernoulli Problem
The St. Petersburg Paradox
Summary Remarks about Mathematical Expectation
Where Do We Eat?
VIII The Law of Averages
The Long Run
Heads or Tails
IX Variability and Chebychev's Theorem
X Binomial Experiments
Pascal's Arithmetic Triangle
Binomial Probability Theorem
Some Characteristics of Binomial Experiments
XI The Law of Large Numbers
Comments About the Classical Law of Large Numbers
Improved Central Limit Theorems
Note on Large Numbers
XII Distribution Functions and Probabilities
The Normal or Gaussian Distribution
What Is Normally Distributed?
"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!"
Note on the Probability of Dealing Any Specified Hand of Thirteen Cards
Further Note on Rare Events
XIV Probability and Statistics
Deduction and Induction
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"
XVI Lady Luck Becomes a Lady
The Probability of an Event
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