WHEN THIS BOOK WAS FIRST CONCEIVED （MORE THAN 25 YEARS AGO）few mathematicians outside the Soviet Union recognized probability as a legitimate branch of mathematics.Applications were limited in scope，and the treatment of individual problems often led to incredible com-plications.Under these circumstances the book could not be written for an existing audience，or to satisfy conscious needs.The hope was rather to attract attention to little-known aspects of probability，to forge links between various parts，to develop unified methods，and to point topotential applications.Because of a growing interest in probability，the book found unexpectedly many users outside mathematical disciplines.Its widespread use was uncterstandable as long as its point of view was new and its material was not otherwise available.But the popularity seems to persist even now，when the contents of most chapters are avail-able in specialized works streamlined for particular needs.For this reason the character of the book remains unchanged in the new edition.I hope that it will continue to serve a variety of needs and，in particular，that it will continue to find readers who read it merely for enjoyment and enlightenment.Throughout the years I was the grateful recipient of many communica-tions from users，and these led to various improvements.Many sections were rewritten to facilitate study.Reading is also improved by a better typeface and the superior editing job by Mrs.H.McDougal： although a professional editor she has preserved a feeling for the requirements of readers and reason.The greatest change is in chapter Ⅲ.This chapter was introduced only in the second edition，which was in fact motivated principally by the unexpected discovery that its enticing material could be treated by elementary methods.But this treatment still depended on combinatorial artifices which have now been replaced by simpler and more natural probabilistic arguments.In essence this new chapter is new.Most conspicuous among other additions are the new sections on branching processes，on Markov chains，and on the De Moivre-Laplace theorem.Chapter ⅩⅧ has been rearranged，and throughout the book there appear minor changes as well as new examples and problems.I regret the misleading nature of the author index，but I felt obliged to state explicitly whenever an idea or example could be traced to a partic-ular source.Unfortunately this means that quotations usually refer to an incidental remark，and are rarely indicative of the nature of the paper quoted.Furthermore，many examples and problems were inspired by reading non-mathematical papers in which related situations are dealt with by different methods.（That newer texts now quote these non-mathematical papers as containing my examples shows how fast prob-ability has developed，but also indicates the limited usefulness of quotations.） Lack of space as well as of competence precluded more adequate historical indications of how probability has changed from thesemimysterious discussions of the twenties to its present flourishing state.For a number of years I have been privileged to work with studentsand younger colleagues to whose help and inspiration I owe much.Much credit for this is due to the support by the U.S.Army Research Office for work in probability at Princeton University.My particularthanks are due to Jay Goldman for a thoughtful memorandum about histeaching experiences，and to Loren Pitt for devoted help with the proofs.
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