second edition adds learning outcomes for each chapter, the R supplements, and many of the chapter review exercises, as well as fixes many typos from the first edition (in both the text and the solutions).
Additional features of the book include the following:
Over 200 examples throughout the text and some 800 end-of-chapter exercises. Includes short numerical solutions for most odd-numbered exercises.
Learning outcomes at the start of each chapter provide information for instructors and students. The learning outcome with a (C) is a computational learning outcome.
End-of-chapter summaries highlight the main ideas and results from each chapter for easy access.
Chapter review exercises, which are provided online, offer a good source of additional problems for students preparing for midterm and/or final exams.
Starred subsections are optional and contain more challenging material and may assume a higher mathematical level.
The R supplements (available online) contain the book code and scripts with enhanced discussion, additional examples, and questions for practice for interested students and instructors.
The introductory R supplement introduces students to the basics of R. (Enhanced version of first edition Appendix A, available online as part of the R supplements.)
A website containing relevant material (including the R supplements, script files, and chapter review exercises) and errata has been established. The URL is www.wiley.com/go/wagaman/probability2e .
An instructor's solutions manual with detailed solutions to all the exercises is available for instructors who teach from this book.
Amy
Amherst, MA
September 2020
ACKNOWLEDGMENTS
From Amy for the second edition:
We are indebted to many individuals who supported the work of creating a second edition of this text. First, we thank Bob, for his thoughts and encouragement when we inquired about revising the text. We also thank our student interns, especially Sabir and Tyler, for their hard work reviewing the text, working on the new supplements, and typing solutions for exercises. The students were generously supported by funding from Amherst College. We also thank our colleagues Nick Horton and Tanya Leise for helpful discussions of the first edition.
Wiley's staff supported us well during the revision, especially Kimberly Monroe-Hill, who had very helpful suggestions. We would also like to thank Mindy Okura-Marszycki, Kathleen Santoloci, and Linda Christina, for their support getting the project off the ground.
From Bob for the first edition:
We are indebted to friends and colleagues who encouraged and supported this project. The students of my Fall 2012 Probability class were real troopers for using an early manuscript that had an embarrassing number of typos and mistakes and offering a volume of excellent advice. We also thank Marty Erickson, Jack Goldfeather, Matthew Rathkey, WenliRui, and Zach Wood-Doughty. Professor Laura Chihara field-tested an early version of the text in her class and has made many helpful suggestions. Thank you to Jack O'Brien at Bowdoin College for a detailed reading of the manuscript and for many suggestions that led to numerous improvements.
Carleton College and the Department of Mathematics were enormously supportive, and I am grateful for a college grant and additional funding that supported this work. Thank you to Mike Tie, the Department's Technical Director, and Sue Jandro, the Department's Administrative Assistant, for help throughout the past year.
The staff at Wiley, including Steve Quigley, Amy Hendrickson, and Sari Friedman, provided encouragement and valuable assistance in preparing this book.
ABOUT THE COMPANION WEBSITE
This book is accompanied by a companion website:
www.wiley.com/go/wagaman/probability2e
The book companion site is split into:
The student companion site includes chapter reviews and is open to all.
The instructor companion site includes the instructor solutions manual.
INTRODUCTION
All theory, dear friend, is gray, but the golden tree of life springs ever green.
—Johann Wolfgang von Goethe
Probability began by first considering games of chance. But today, it has practical applications in areas as diverse as astronomy, economics, social networks, and zoology that enrich the theory and give the subject its unique appeal.
In this book, we will flip coins, roll dice, and pick balls from urns, all the standard fare of a probability course. But we have also tried to make connections with real-life applications and illustrate the theory with examples that are current and engaging.
You will see some of the following case studies again throughout the text. They are meant to whet your appetite for what is to come.
I.1 Walking the Web
There are about one trillion websites on the Internet. When you google a phrase like “Can Chuck Norris divide by zero?,” a remarkable algorithm called PageRank searches these sites and returns a list ranked by importance and relevance, all in the blink of an eye. PageRank is the heart of the Google search engine. The algorithm assigns an “importance value” to each web page and gives it a rank to determine how useful it is.
PageRank is a significant accomplishment of mathematics and linear algebra. It can be understood using probability. Of use are probability concepts called Markov chains and random walks, explored in Chapter 11. Imagine a web surfer who starts at some web page and clicks on a link at random to find a new site. At each page, the surfer chooses from one of the available hypertext links equally at random. If there are two links, it is a coin toss, heads or tails, to decide which one to pick. If there are 100 links, each one has a 1% chance of being chosen. As the web surfer moves from page to random page, they are performing a random walk on the web.
What is the PageRank of site
The PageRank algorithm is actually best understood as an assignment of probabilities to each site on the web. Such a list of numbers is called a probability distribution. And since it comes as the result of a theoretically infinitely long random walk, it is known as the limiting distribution of the random walk. Remarkably, the PageRank values for billions of websites can be computed quickly and in real time.
I.2 Benford's Law
Turn to a random page in this book. Look in the middle of the page and point to the first number you see. Write down the first digit of that number.
You might think that such first digits are equally likely to be any integer from 1 to 9. But a remarkable probability