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\markboth{PREFACE}{PREFACE} 
Probability theory began in seventeenth century France when the two great
French 
mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two
problems from 
games of chance. Problems like those Pascal and Fermat solved continued to
influence such 
early researchers as Huygens, Bernoulli, and DeMoivre in establishing a
mathematical theory of 
probability. Today, probability theory is a well-established branch of
mathematics that finds 
applications in every area of scholarly activity from music to physics, and in
daily experience 
from weather prediction to predicting the risks of new medical treatments.
\par
This text is designed for an introductory probability course taken by
sophomores, juniors, and
seniors in mathematics, the physical and social sciences, engineering, and
computer
science. It  presents a thorough treatment of probability ideas and techniques
necessary for a firm 
understanding of the subject. The text can be used in a variety of course
lengths, levels, and areas 
of emphasis.
\par
For use in a standard one-term course, in which both discrete and continuous
probability is covered,
students should have taken as a prerequisite two terms of calculus, including
an introduction to
multiple integrals. In order to cover Chapter~\ref{chp 11}, which contains
material on Markov
chains,  some knowledge of matrix theory is necessary.
\par
The text can also be used in a discrete probability course. The material has
been organized in 
such a way that the discrete and continuous probability discussions are
presented in a separate, 
but parallel, manner. This organization dispels an overly rigorous or formal
view of probability 
and offers some strong pedagogical value in that the discrete discussions can
sometimes serve to 
motivate the more abstract continuous probability discussions. For use in a
discrete probability 
course, students should have taken one term of calculus as a prerequisite.
\par
Very little computing background is assumed or necessary in order to obtain
full benefits from the use of the computing material and examples in the text.  All of the
programs that are used in the text have been written in each of
the languages TrueBASIC, Maple, and Mathematica.   
\par
This book is on the Web at http://www.dartmouth.edu/$\tilde{\ }$chance, and is part
of the Chance project, which is devoted to providing materials for beginning
courses in probability
and statistics.  The computer programs, solutions to the odd-numbered
exercises, and current errata
are also available at this site.  Instructors may obtain all of the solutions
by writing to either
of the authors, at jlsnell@dartmouth.edu and cgrinst1@swarthmore.edu.  It is
our intention to place
items related to this book at this site, and we invite our readers to submit
their contributions.
\par
\bigskip\centerline{\bf FEATURES}\medskip
\par
{\it Level of rigor and emphasis:} Probability is a wonderfully intuitive and
applicable field of 
mathematics. We have tried not to spoil its beauty by presenting too much
formal mathematics. 
Rather, we have tried to develop the key ideas in a somewhat leisurely style,
to provide a variety 
of interesting applications to probability, and to show some of the
nonintuitive examples that 
make probability such a lively subject.

{\it Exercises:} There are over 600 exercises in the text providing plenty of
opportunity for practicing 
skills and developing a sound understanding of the ideas. In the exercise sets
are routine 
exercises to be done with and without the use of a computer and more
theoretical exercises to 
improve the understanding of basic concepts. More difficult exercises are
indicated by an 
asterisk.  A solution manual for all of the exercises is available to
instructors.

{\it Historical remarks:} Introductory probability is a subject in which the
fundamental ideas are still 
closely tied to those of the founders of the subject. For this reason, there
are numerous historical 
comments in the text, especially as they deal with the development of discrete
probability.

{\it Pedagogical use of computer programs:} Probability theory makes
predictions about experiments 
whose outcomes depend upon chance. Consequently, it lends itself beautifully to
the use of 
computers as a mathematical tool to simulate and analyze chance experiments.

In the text the computer is utilized in several ways. First, it provides a
laboratory where chance 
experiments can be simulated and the students can get a feeling for the variety
of such 
experiments. This use of the computer in probability has been already
beautifully illustrated by 
William Feller in the second edition of his famous text {\it An Introduction to
Probability Theory and Its 
Applications} (New York: Wiley, 1950). In the preface, Feller wrote about his
treatment of 
fluctuation in coin tossing: ``The results are so amazing and so at variance
with common intuition 
that even sophisticated colleagues doubted that coins actually misbehave as
theory predicts. The 
record of a simulated experiment is therefore included."
\par
In addition to providing a laboratory for the student, the computer is a
powerful aid in 
understanding basic results of probability theory. For example, the graphical
illustration of the 
approximation of the standardized binomial distributions to the normal curve is a
more convincing 
demonstration of the Central Limit Theorem than many of the formal proofs of
this fundamental 
result.
\par
Finally, the computer allows the student to solve problems that do not lend
themselves to 
closed-form formulas such as waiting times in queues. Indeed, the introduction
of the computer 
changes the way in which we look at many problems in probability. For example,
being able to 
calculate exact binomial probabilities for experiments up to 1000 trials
changes the way we view 
the normal and Poisson approximations.
\par
\pagebreak[4]
\bigskip\centerline{\bf ACKNOWLEDGMENTS FOR FIRST EDITION}\medskip
\par
Anyone writing a probability text today owes a great debt to William Feller,
who taught us all how to make probability come alive as a subject matter.  If
you find an example, an application, or an exercise that you really like, it
probably had its origin in Feller's classic text, {\sl An Introduction to
Probability Theory and Its Applications.}

This book had its start with a course given jointly at Dartmouth College with
Professor John Kemeny.  I am indebted to Professor Kemeny for convincing me
that it is both useful and fun to use the computer in the study of
probability.  He has continuously and generously shared his ideas on
probability and computing with me.  No less impressive has been the help of
John Finn in making the computing an integral part of the text and in writing
the programs so that they not only can be easily used, but they also can be
understood and modified by the student to explore further problems.  Some of
the programs in the text were developed through collaborative efforts with John
Kemeny and Thomas Kurtz on a Sloan Foundation project and with John Finn on a
Keck Foundation project.  I am grateful to both foundations for their support.

I am indebted to many other colleagues, students, and friends for valuable
comments and suggestions.  A few whose names stand out are: Eric and Jim
Baumgartner, Tom Bickel, Bob Beck, Ed Brown, Christine Burnley, Richard
Crowell, David Griffeath, John Lamperti, Beverly Nickerson, Reese Prosser,
Cathy Smith, and Chris Thron.

The following individuals were kind enough to review various drafts of the
manuscript.  Their encouragement, criticisms, and suggestions were very
helpful.
\vskip .2in
\begin{tabular}{ll}
{Ron Barnes} & {\sl University of Houston, Downtown College} \\
{Thomas Fischer} & {\sl Texas A \& M University} \\
{Richard Groeneveld} & {\sl Iowa State University} \\
{James Kuelbs} & {\sl University of Wisconsin, Madison} \\
{Greg Lawler} & {\sl Duke University} \\
{Sidney Resnick} & {\sl Colorado State University} \\
{Malcom Sherman} & {\sl SUNY Albany} \\
{Olaf Stackelberg} & {\sl Kent State University} \\
{Murad Taqqu} & {\sl Boston University} \\
{Abraham Wender} & {\sl University of North Carolina}\\
\end{tabular}
\vskip .2in

In addition, I would especially like to thank James Kuelbs, Sidney Resnick, and
their students for using the manuscript in their courses and sharing their
experience and invaluable suggestions with me.

The versatility of Dartmouth's mathematical word processor PREPPY, written by
Professor James Baumgartner, has made it much easier to make revisions, but has
made the job of typist extraordinaire Marie Slack correspondingly more
challenging.  Her high standards and willingness always to try the next more
difficult task have made it all possible.

Finally, I must thank all the people at Random House who helped during the
development and production of this project.  First, among these was my editor
Wayne Yuhasz, whose continued encouragement and commitment were very helpful
during the development of the manuscript.  The entire production team provided
efficient and professional support: Margaret Pinette, project manager; Michael
Weinstein, production manager; and Kate Bradfor of Editing, Design, and
Production, Inc.
\par
\bigskip\centerline{\bf ACKNOWLEDGMENTS FOR SECOND EDITION}\medskip
\par
The debt to William Feller has not diminished in the years between the two
editions of this book. 
His book on probability is likely to remain the classic book in this field for
many years.
\par
The process of revising the first edition of this book began with some
high-level discussions involving the two present co-authors together with Reese Prosser and
John Finn.  It was during these discussions that, among other things, the first co-author was
made aware of the concept of ``negative royalties" by Professor Prosser.  
\par
We are indebted to many people for their help in this undertaking.  First and
foremost, we thank
Mark Kernighan for his almost 40 pages of single-spaced comments on the first
edition.  Many of
these comments were very thought-provoking; in addition, they provided a
student's perspective on the
book.  Most of the major changes in the second edition have their genesis in
these notes.
\par
We would also like to thank Fuxing Hou, who provided extensive help with the
typesetting and the
figures.  Her incessant good humor in the face of many trials, both big (``we
need to change the
entire book from Lamstex to Latex") and small (``could you please move this
subscript down just a
bit?"), was truly remarkable.
\par
We would also like to thank Lee Nave, who typed the entire first edition of the
book into the
computer.  Lee corrected most of the typographical errors in the first edition during
this process, making our job easier.
\par
Karl Knaub and Jessica Sklar are responsible for the implementations of the
computer programs in
Mathematica and Maple, and we thank them for their efforts.  We
also thank Jessica for her work on the solution manual for the exercises,
building on the work done
by Gang Wang for the first edition.
\par
Tom Shemanske and Dana Williams provided much TeX-nical assistance.  Their
patience and willingness
to help, even to the extent of writing intricate TeX macros, are very much
appreciated.
\par
The following people used various versions of the second edition in their
probability courses, and
provided valuable comments and criticisms.
\vskip .2in
\begin{tabular}{ll}
{Marty Arkowitz} & {\sl Dartmouth College}\\
{Aimee Johnson} & {\sl Swarthmore College}\\
{Bill Peterson} & {\sl Middlebury College}\\
{Dan Rockmore} & {\sl Dartmouth College}\\
{Shunhui Zhu} & {\sl Dartmouth College}\\
\end{tabular}
\vskip .2in
Reese Prosser and John Finn provided much in the way of moral support and
camaraderie throughout this
project.  Certainly, one of the high points of this entire endeavour was
Professor Prosser's
telephone call to a casino in Monte Carlo, in an attempt to find out the rules
involving the
``prison" in roulette.
\par
Peter Doyle motivated us to make this book
part of a larger project on the Web, to which others can contribute.  He also
spent many hours
actually carrying out the operation of putting the book on the Web.
\par
Finally, we thank Sergei Gelfand and the American Mathematical Society for
their interest in our
book, their help in its production, and their willingness to let us put the
book on the Web.

