The Probability Lifesaver: All the Tools You Need to Understand Chance
The essential lifesaver for students who want to master probability

Series:
 Princeton Lifesaver Study Guides
Hardcover
 Price:
 $99.95 / £78.00
 ISBN:
 Published:
 May 16, 2017
 Copyright:
 2017
 Pages:
 752
 Size:
 7 x 10 in.
 Illus:
 8 color illus. 64 line illus. 21 tables.
Paperback
 Price:
 $99.95 / £78.00
 ISBN:
 Published:
 May 16, 2017
 Copyright:
 2017
 Pages:
 752
 Size:
 7 x 10 in.
 Illus:
 8 color illus. 64 line illus. 21 tables.
ebook
 Price:
 $99.95 / £78.00
 ISBN:
 Published:
 May 16, 2017
 Copyright:
 2017
 Pages:
 752
 Size:
 7 x 10 in.
 Illus:
 8 color illus. 64 line illus. 21 tables.
For students learning probability, its numerous applications, techniques, and methods can seem intimidating and overwhelming. That’s where The Probability Lifesaver steps in. Designed to serve as a complete standalone introduction to the subject or as a supplement for a course, this accessible and userfriendly study guide helps students comfortably navigate probability’s terrain and achieve positive results.
The Probability Lifesaver is based on a successful course that Steven Miller has taught at Brown University, Mount Holyoke College, and Williams College. With a relaxed and informal style, Miller presents the math with thorough reviews of prerequisite materials, workedout problems of varying difficulty, and proofs. He explores a topic first to build intuition, and only after that does he dive into technical details. Coverage of topics is comprehensive, and materials are repeated for reinforcement—both in the guide and on the book’s website. An appendix goes over proof techniques, and video lectures of the course are available online. Students using this book should have some familiarity with algebra and precalculus.
The Probability Lifesaver not only enables students to survive probability but also to achieve mastery of the subject for use in future courses.
 A helpful introduction to probability or a perfect supplement for a course
 Numerous workedout examples
 Lectures based on the chapters are available free online
 Intuition of problems emphasized first, then technical proofs given
 Appendixes review proof techniques
 Relaxed, conversational approach
 Note to Readers
 How to Use This Book
 I General Theory
 1 Introduction
 1.1 Birthday Problem
 1.1.1 Stating the Problem
 1.1.2 Solving the Problem
 1.1.3 Generalizing the Problem and Solution: Efficiencies
 1.1.4 Numerical Test
 1.2 From Shooting Hoops to the Geometric Series
 1.2.1 The Problem and Its Solution
 1.2.2 Related Problems
 1.2.3 General Problem Solving Tips
 1.3 Gambling
 1.3.1 The 2008 Super Bowl Wager
 1.3.2 Expected Returns
 1.3.3 The Value of Hedging
 1.3.4 Consequences
 1.4 Summary
 1.5 Exercises
 1.1 Birthday Problem
 2 Basic Probability Laws
 2.1 Paradoxes
 2.2 Set Theory Review
 2.2.1 Coding Digression
 2.2.2 Sizes of Infinity and Probabilities
 2.2.3 Open and Closed Sets
 2.3 Outcome Spaces, Events, and the Axioms of Probability
 2.4 Axioms of Probability
 2.5 Basic Probability Rules
 2.5.1 Law of Total Probability
 2.5.2 Probabilities of Unions
 2.5.3 Probabilities of Inclusions
 2.6 Probability Spaces and σalgebras
 2.7 Appendix: Experimentally Finding Formulas
 2.7.1 Product Rule for Derivatives
 2.7.2 Probability of a Union
 2.8 Summary
 2.9 Exercises
 3 Counting I: Cards
 3.1 Factorials and Binomial Coefficients
 3.1.1 The Factorial Function
 3.1.2 Binomial Coefficients
 3.1.3 Summary
 3.2 Poker
 3.2.1 Rules
 3.2.2 Nothing
 3.2.3 Pair
 3.2.4 Two Pair
 3.2.5 Three of a Kind
 3.2.6 Straights, Flushes, and Straight Flushes
 3.2.7 Full House and Four of a Kind
 3.2.8 Practice Poker Hand: I
 3.2.9 Practice Poker Hand: II
 3.3 Solitaire
 3.3.1 Klondike
 3.3.2 Aces Up
 3.3.3 FreeCell
 3.4 Bridge
 3.4.1 Tictactoe
 3.4.2 Number of Bridge Deals
 3.4.3 Trump Splits
 3.5 Appendix: Coding to Compute Probabilities
 3.5.1 Trump Split and Code
 3.5.2 Poker Hand Codes
 3.6 Summary
 3.7 Exercises
 3.1 Factorials and Binomial Coefficients
 4 Conditional Probability, Independence, and Bayes’ Theorem
 4.1 Conditional Probabilities
 4.1.1 Guessing the Conditional Probability Formula
 4.1.2 Expected Counts Approach
 4.1.3 Venn Diagram Approach
 4.1.4 The Monty Hall Problem
 4.2 The General Multiplication Rule
 4.2.1 Statement
 4.2.2 Poker Example
 4.2.3 Hat Problem and Error Correcting Codes
 4.2.4 Advanced Remark: Definition of Conditional Probability
 4.3 Independence
 4.4 Bayes’ Theorem
 4.5 Partitions and the Law of Total Probability
 4.6 Bayes’ Theorem Revisited
 4.7 Summary
 4.8 Exercises
 4.1 Conditional Probabilities
 5 Counting II: InclusionExclusion
 5.1 Factorial and Binomial Problems
 5.1.1 “How many” versus “What’s the probability”
 5.1.2 Choosing Groups
 5.1.3 Circular Orderings
 5.1.4 Choosing Ensembles
 5.2 The Method of InclusionExclusion
 5.2.1 Special Cases of the InclusionExclusion Principle
 5.2.2 Statement of the InclusionExclusion Principle
 5.2.3 Justification of the InclusionExclusion Formula
 5.2.4 Using InclusionExclusion: Suited Hand
 5.2.5 The At Least to Exactly Method
 5.3 Derangements
 5.3.1 Counting Derangements
 5.3.2 The Probability of a Derangement
 5.3.3 Coding Derangement Experiments
 5.3.4 Applications of Derangements
 5.4 Summary
 5.5 Exercises
 5.1 Factorial and Binomial Problems
 6 Counting III: Advanced Combinatorics
 6.1 Basic Counting
 6.1.1 Enumerating Cases: I
 6.1.2 Enumerating Cases: II
 6.1.3 Sampling With and Without Replacement
 6.2 Word Orderings
 6.2.1 Counting Orderings
 6.2.2 Multinomial Coefficients
 6.3 Partitions
 6.3.1 The Cookie Problem
 6.3.2 Lotteries
 6.3.3 Additional Partitions
 6.4 Summary
 6.5 Exercises
 6.1 Basic Counting
 1 Introduction
 II Introduction to Random Variables
 7 Introduction to Discrete Random Variables
 7.1 Discrete Random Variables: Definition
 7.2 Discrete Random Variables: PDFs
 7.3 Discrete Random Variables: CDFs
 7.4 Summary
 7.5 Exercises
 8 Introduction to Continuous Random Variables
 8.1 Fundamental Theorem of Calculus
 8.2 PDFs and CDFs: Definitions
 8.3 PDFs and CDFs: Examples
 8.4 Probabilities of Singleton Events
 8.5 Summary
 8.6 Exercises
 9 Tools: Expectation
 9.1 Calculus Motivation
 9.2 Expected Values and Moments
 9.3 Mean and Variance
 9.4 Joint Distributions
 9.5 Linearity of Expectation
 9.6 Properties of the Mean and the Variance
 9.7 Skewness and Kurtosis
 9.8 Covariances
 9.9 Summary
 9.10 Exercises
 10 Tools: Convolutions and Changing Variables
 10.1 Convolutions: Definitions and Properties
 10.2 Convolutions: Die Example
 10.2.1 Theoretical Calculation
 10.2.2 Convolution Code
 10.3 Convolutions of Several Variables
 10.4 Change of Variable Formula: Statement
 10.5 Change of Variables Formula: Proof
 10.6 Appendix: Products and Quotients of Random Variables
 10.6.1 Density of a Product
 10.6.2 Density of a Quotient
 10.6.3 Example: Quotient of Exponentials
 10.7 Summary
 10.8 Exercises
 11 Tools: Differentiating Identities
 11.1 Geometric Series Example
 11.2 Method of Differentiating Identities
 11.3 Applications to Binomial Random Variables
 11.4 Applications to Normal Random Variables
 11.5 Applications to Exponential Random Variables
 11.6 Summary
 11.7 Exercises
 7 Introduction to Discrete Random Variables
 III Special Distributions
 12 Discrete Distributions
 12.1 The Bernoulli Distribution
 12.2 The Binomial Distribution
 12.3 The Multinomial Distribution
 12.4 The Geometric Distribution
 12.5 The Negative Binomial Distribution
 12.6 The Poisson Distribution
 12.7 The Discrete Uniform Distribution
 12.8 Exercises
 13 Continuous Random Variables: Uniform and Exponential
 13.1 The Uniform Distribution
 13.1.1 Mean and Variance
 13.1.2 Sums of Uniform Random Variables
 13.1.3 Examples
 13.1.4 Generating Random Numbers Uniformly
 13.2 The Exponential Distribution
 13.2.1 Mean and Variance
 13.2.2 Sums of Exponential Random Variables
 13.2.3 Examples and Applications of Exponential Random Variables
 13.2.4 Generating Random Numbers from Exponential Distributions
 13.3 Exercises
 13.1 The Uniform Distribution
 14 Continuous Random Variables: The Normal Distribution
 14.1 Determining the Normalization Constant
 14.2 Mean and Variance
 14.3 Sums of Normal Random Variables
 14.3.1 Case 1: μX = μY = 0 and 2/x = 2/y = 1
 14.3.2 Case 2: General μX, μY and 2/x, 2/y
 14.3.3 Sums of Two Normals: Faster Algebra
 14.4 Generating Random Numbers from Normal Distributions
 14.5 Examples and the Central Limit Theorem
 14.6 Exercises
 15 The Gamma Function and Related Distributions
 15.1 Existence of Γ (s)
 15.2 The Functional Equation of Γ (s)
 15.3 The Factorial Function and Γ (s)
 15.4 Special Values of Γ (s)
 15.5 The Beta Function and the Gamma Function
 15.5.1 Proof of the Fundamental Relation
 15.5.2 The Fundamental Relation and Γ(1/2)
 15.6 The Normal Distribution and the Gamma Function
 15.7 Families of Random Variables
 15.8 Appendix: Cosecant Identity Proofs
 15.8.1 The Cosecant Identity: First Proof
 15.8.2 The Cosecant Identity: Second Proof
 15.8.3 The Cosecant Identity: Special Case s = 1/2
 15.9 Cauchy Distribution
 15.10 Exercises
 16 The Chisquare Distribution
 16.1 Origin of the Chisquare Distribution
 16.2 Mean and Variance of X ∼ χ2(1)
 16.3 Chisquare Distributions and Sums of Normal Random Variables
 16.3.1 Sums of Squares by Direct Integration
 16.3.2 Sums of Squares by the Change of Variables Theorem
 16.3.3 Sums of Squares by Convolution
 16.3.4 Sums of Chisquare Random Variables
 16.4 Summary
 16.5 Exercises
 12 Discrete Distributions
 IV Limit Theorems
 17 Inequalities and Laws of Large Numbers
 17.1 Inequalities
 17.2 Markov’s Inequality
 17.3 Chebyshev’s Inequality
 17.3.1 Statement
 17.3.2 Proof
 17.3.3 Normal and Uniform Examples
 17.3.4 Exponential Example
 17.4 The Boole and Bonferroni Inequalities
 17.5 Types of Convergence
 17.5.1 Convergence in Distribution
 17.5.2 Convergence in Probability
 17.5.3 Almost Sure and Sure Convergence
 17.6 Weak and Strong Laws of Large Numbers
 17.7 Exercises
 18 Stirling’s Formula
 18.1 Stirling’s Formula and Probabilities
 18.2 Stirling’s Formula and Convergence of Series
 18.3 From Stirling to the Central Limit Theorem
 18.4 Integral Test and the Poor Man’s Stirling
 18.5 Elementary Approaches towards Stirling’s Formula
 18.5.1 Dyadic Decompositions
 18.5.2 Lower Bounds towards Stirling: I
 18.5.3 Lower Bounds toward Stirling II
 18.5.4 Lower Bounds towards Stirling: III
 18.6 Stationary Phase and Stirling
 18.7 The Central Limit Theorem and Stirling
 18.8 Exercises
 19 Generating Functions and Convolutions
 19.1 Motivation
 19.2 Definition
 19.3 Uniqueness and Convergence of Generating Functions
 19.4 Convolutions I: Discrete Random Variables
 19.5 Convolutions II: Continuous Random Variables
 19.6 Definition and Properties of Moment Generating Functions
 19.7 Applications of Moment Generating Functions
 19.8 Exercises
 20 Proof of the Central Limit Theorem
 20.1 Key Ideas of the Proof
 20.2 Statement of the Central Limit Theorem
 20.3 Means, Variances, and Standard Deviations
 20.4 Standardization
 20.5 Needed Moment Generating Function Results
 20.6 Special Case: Sums of Poisson Random Variables
 20.7 Proof of the CLT for General Sums via MGF
 20.8 Using the Central Limit Theorem
 20.9 The Central Limit Theorem and Monte Carlo Integration
 20.10 Summary
 20.11 Exercises
 21 Fourier Analysis and the Central Limit Theorem
 21.1 Integral Transforms
 21.2 Convolutions and Probability Theory
 21.3 Proof of the Central Limit Theorem
 21.4 Summary
 21.5 Exercises
 17 Inequalities and Laws of Large Numbers
 V Additional Topics
 22 Hypothesis Testing
 22.1 Ztests
 22.1.1 Null and Alternative Hypotheses
 22.1.2 Significance Levels
 22.1.3 Test Statistics
 22.1.4 Onesided versus Twosided Tests
 22.2 On pvalues
 22.2.1 Extraordinary Claims and pvalues
 22.2.2 Large pvalues
 22.2.3 Misconceptions about pvalues
 22.3 On ttests
 22.3.1 Estimating the Sample Variance
 22.3.2 From ztests to ttests
 22.4 Problems with Hypothesis Testing
 22.4.1 Type I Errors
 22.4.2 Type II Errors
 22.4.3 Error Rates and the Justice System
 22.4.4 Power
 22.4.5 Effect Size
 22.5 Chisquare Distributions, Goodness of Fit
 22.5.1 Chisquare Distributions and Tests of Variance
 22.5.2 Chisquare Distributions and tdistributions
 22.5.3 Goodness of Fit for List Data
 22.6 Two Sample Tests
 22.6.1 Twosample ztest: Known Variances
 22.6.2 Twosample ttest: Unknown but Same Variances
 22.6.3 Unknown and Different Variances
 22.7 Summary
 22.8 Exercises
 22.1 Ztests
 23 Difference Equations, Markov Processes, and Probability
 23.1 From the Fibonacci Numbers to Roulette
 23.1.1 The Doubleplusone Strategy
 23.1.2 A Quick Review of the Fibonacci Numbers
 23.1.3 Recurrence Relations and Probability
 23.1.4 Discussion and Generalizations
 23.1.5 Code for Roulette Problem
 23.2 General Theory of Recurrence Relations
 23.2.1 Notation
 23.2.2 The Characteristic Equation
 23.2.3 The Initial Conditions
 23.2.4 Proof that Distinct Roots Imply Invertibility
 23.3 Markov Processes
 23.3.1 Recurrence Relations and Population Dynamics
 23.3.2 General Markov Processes
 23.4 Summary
 23.5 Exercises
 23.1 From the Fibonacci Numbers to Roulette
 24 The Method of Least Squares
 24.1 Description of the Problem
 24.2 Probability and Statistics Review
 24.3 The Method of Least Squares
 24.4 Exercises
 25 Two Famous Problems and Some Coding
 25.1 The Marriage/Secretary Problem
 25.1.1 Assumptions and Strategy
 25.1.2 Probability of Success
 25.1.3 Coding the Secretary Problem
 25.2 Monty Hall Problem
 25.2.1 A Simple Solution
 25.2.2 An Extreme Case
 25.2.3 Coding the Monty Hall Problem
 25.3 Two Random Programs
 25.3.1 Sampling with and without Replacement
 25.3.2 Expectation
 25.4 Exercises
 25.1 The Marriage/Secretary Problem
 Appendix A Proof Techniques
 A.1 How to Read a Proof
 A.2 Proofs by Induction
 A.2.1 Sums of Integers
 A.2.2 Divisibility
 A.2.3 The Binomial Theorem
 A.2.4 Fibonacci Numbers Modulo 2
 A.2.5 False Proofs by Induction
 A.3 Proof by Grouping
 A.4 Proof by Exploiting Symmetries
 A.5 Proof by Brute Force
 A.6 Proof by Comparison or Story
 A.7 Proof by Contradiction
 A.8 Proof by Exhaustion (or Divide and Conquer)
 A.9 Proof by Counterexample
 A.10 Proof by Generalizing Example
 A.11 Dirichlet’s PigeonHole Principle
 A.12 Proof by Adding Zero or Multiplying by One
 Appendix B Analysis Results
 B.1 The Intermediate and Mean Value Theorems
 B.2 Interchanging Limits, Derivatives, and Integrals
 B.2.1 Interchanging Orders: Theorems
 B.2.2 Interchanging Orders: Examples
 B.3 Convergence Tests for Series
 B.4 BigOh Notation
 B.5 The Exponential Function
 B.6 Proof of the CauchySchwarz Inequality
 B.7 Exercises
 Appendix C Countable and Uncountable Sets
 C.1 Sizes of Sets
 C.2 Countable Sets
 C.3 Uncountable Sets
 C.4 Length of the Rationals
 C.5 Length of the Cantor Set
 C.6 Exercises
 Appendix D Complex Analysis and the Central Limit Theorem
 D.1 Warnings from Real Analysis
 D.2 Complex Analysis and Topology Definitions
 D.3 Complex Analysis and Moment Generating Functions
 D.4 Exercises
 22 Hypothesis Testing
 Bibliography
 Index
"I recommend the book to everyone who is studying and fascinated by statistics."—Singalakha Menziwa, Mathemafrica
"Steven J. Miller’s The Probability Lifesaver presents, as its subtitle claims, 'all the tools you need to understand chance' in a clear, straightforward manner. . . . For the students that have a good understanding of Calculus, the combination of the probability discussions along with the calculus behind these topics is very beneficial."—MAA Reviews
"The breadth of the book’s coverage and its clear, informal tone in addressing highly formal problems remind one of a friendly professor offering unlimited office hours, and the book will be a highly accessible supplement for students working through another, more conventional text. . . . [This is] a volume that deserves to be widely known in educational circles and will likely find its way to the shelves of practicing statisticians who wish to probe below the surface of fundamental theorems that they have learned by rote."—H. Van Dyke Parunak, Computing Reviews
"This is a superb book by a gifted writer and mathematician. Miller's amiable, intuitive writing style weaves stories about probability into the narrative in a unique fashion."—Larry Leemis, College of William & Mary
"The Probability Lifesaver creates a wonderful mathematical experience. It combines important theories with fun problems, giving a new and creative perspective on probability. This book helped me understand the big questions behind the mathematics of probability: why the complex theories I was learning are true, where they come from, and what are their applications. This approach is a welcome complement to other heavy theoretical books, and was detailed and expansive enough to serve as the main textbook for our class."—Alexandre Gueganic, Williams College ‘19
"This fun book gives readers the feeling that they are having a live conversation with the author. A wonderful resource for students and teachers alike, The Probability Lifesaver contains clear and detailed explanations, problems with solutions on every topic, and extremely helpful background material."—Iddo BenAri, University of Connecticut
"In The Probability Lifesaver, Miller does more than simply present the theoretical framework of probability. He takes complex concepts and describes them in understandable language, provides realistic applications that highlight the farextending reaches of probability, and engages the problemsolving intuitions that lie at the heart of mathematics. Lastly, and most importantly, I am reminded throughout this textbook of why I chose to study mathematics: because it's fun!"—Michael Stone, Williams College ‘16
"The Probability Lifesaver motivates introductory probability theory with concrete applications in an approachable and engaging manner. From computing the probability of various poker hands to defining sigmaalgebras, it strikes a balance between applied computation and mathematical theory that makes it easy to follow while still being mathematically satisfying."—David Burt, Williams College ‘17
"A balanced mix of theoretical and practical problemsolving approaches in probability—suited for personal study as well as textbook reading in and out of the classroom. After college, while working, I took a probability class remotely and with this book, I was able to follow easily despite being without a TA or easy access to the professor. From research examples to interview questions, it has saved my life more than once."—Dan Zhao, Williams College ‘ 14
"The Probability Lifesaver helped me build a foundation of probability theory and an appreciation for its nuances through engaging examples and easytofollow explanations. This wellwritten and extensive book will serve as your guide to probability and reward you for the time you give it."—Jaclyn Porfilio, Williams '15
"I see a tremendous value in this fun, engaging, and informal book. It has a conversational tone, which invites students to engage the material and concepts. It is as if Miller is there, lecturing on the topics, helping students to think things through for themselves."—John Imbrie, University of Virginia
"The Probability Lifesaver contains a lot of explanations and examples and provides stepbystep instructions to how definitions and ideas are formulated. I appreciated that it tries to provide multiple solutions to each problem. Interesting, informative, approachable, and comprehensive, this book was easy to read and would make a good supplement for a first probability course at the undergraduate level."—Jingchen Hu, Vassar College
"Filled with many interesting and contemporary examples, The Probability Lifesaver would have undoubtedly helped me while I was taking statistics. Miller offers careful, detailed explanations in simple terms that are easy to understand."—James Coyle, former student at Rutgers University