2.1. Review by: D R Cox.
Journal of the Royal Statistical Society Series A (General) 121 (3) (1958), 354-355.

The first edition of this outstanding book appeared in 1950 and it is now firmly established both as an introductory textbook and as a reference work on more advanced topics. For the second edition Professor Feller has put in a new chapter, carried out extensive minor revisions and added a considerable number of further exercises.

2.2. Review by: Boyd Harshbarger.
American Institute of Biological Sciences Bulletin 8 (1) (1958), 32.

This book represents an enlargement of the older work rather than a revision. The first edition has been brought up to date and enlarged by including additional material on the power of combinatorial methods, recurrent events, and general fluctuation theory. The book is restricted to discrete sample space so that the basic concepts of probability theory can be presented from an intuitive background. ... In general, the most elementary part of the book will be too advanced for the average biological student

2.3. Review by: J Wolfowitz.
Management Science 5 (3) (1959), 336-337.

This is the second edition of a book which first appeared in 1950 and received an enthusiastic reception. In the opinion of this reviewer that reception was well merited; seldom has he read a mathematical book so charming and so valuable. It takes a reader, who need have no previous acquaintance with probability, but who possesses a moderate amount of mathematical sophistication, well into many parts of the subject, and fits him to pursue further studies in the subject. Not least of its charms is the abundance of interesting, natural, and important applications of the subject to many branches of science. The applications explain many of the directions in which the theory of probability developed, and the application of probability illuminates the physical situations discussed. This interplay of theory and application, which occurs in other branches of mathematics and is so marked in probability theory, is given ample expression in this book, to the enhancement of both. This is done while maintaining a high level of mathematical rigour and development. ... It goes without saying that this book, with its wealth of information and charm, is indispensable for the library of any worker in management science who has the mathematical competence to read it.

2.4. Review by: F G Foster.
Journal of the Royal Statistical Society Series C (Applied Statistics) 7 (3) (1958), 203-204.

The first edition of this book, which appeared in 1950, quickly became established as the standard reference and textbook in probability theory, and has since played a key role in stimulating research in this subject. The new edition follows the same general pattern as the old, and, at first glance, appears to be not much altered. In fact it has throughout been fairly extensively re-written and streamlined, and the type has been reset. This has enabled Professor Feller to pack additional material into a space which readers of the first edition will recall was fairly tightly crammed with ideas.

2.5. Review by: Paul H Randolph.
Operations Research 6 (4) (1958), 636-637.

No doubt a large number of operations researchers were disappointed with IN the announcement last September of the publication of the second edition of Feller's 'Probability, Volume One'. Since the first edition of Volume One has been highly respected by almost every operations researcher, many have long been eagerly waiting for Volume Two rather than this revised edition. Thus the question of the need of the second edition arises quite naturally. That is, is the second edition a significant improvement over the first? At first glance it appears that there may have been only few changes in the revision. There are still seventeen chapters plus the introduction. These begin with the sample-space discussion and end with the familiar chapter on queuing. Some of the intervening chapters look very much like corresponding chapters in the first edition, and upon closer examination are. Thus, the general plan of the second edition is approximately the same as the first. However, there do exist distinct differences in this second edition. It does not take long to discover a definite streamlining in most chapters. In addition, the elements of queuing theory have been woven into the context throughout the book and serve 'as a unifying thread.' ... For those who know the first edition, it is no surprise to note that this is an extremely well-written text. The general streamlining has done nothing to spoil the readability of the book, but on the contrary has improved it.

2.6. Review by: Jerome Rothstein.
Science, New Series 129 (3354) (1959), 956.

The first edition of this book was very successful, and this second edition should be even more so. Most books which attempt to develop probability theory rigorously are readable from the mathematician's standpoint but assume too much familiarity with recondite branches of mathematics to attract most physicists, engineers, or others needing probability and statistics as tools. They also tend to present abstract discussions largely devoid of applications. The present book is rigorous but contains a wealth of illustrative material and examples relative to physics, genetics, contagious disease, card games, traffic and queuing problems, industrial quality control, chain reactions, engineering, and statistics. Most of the chapters, include numerous problems, ranging from simple exercises to applications and extensions of the text.

2.7. Review by: W A W.
Journal of the American Statistical Association 53 (281) (1958), 215.

In a Preface to this second edition , Feller explains that numerous communications from readers of the first edition stimulated him "to think of improvements and to collect better examples and exercises," which "will make for easier reading and teaching from the book."

2.8. Review by: M E Munroe.
The American Mathematical Monthly 65 (7) (1958), 538.

Feller has long been interested in the probable vagaries of a single sequence of trials as contrasted with the mean behaviour of an aggregate of sequences. In this connection he has inserted a new chapter (Chapter III) in which he gives a very elegant, yet elementary, derivation of some rather startling theorems on coin tossing. The major change is that the theory of recurrent events has been pushed forwards so as permeate the entire book, whereas in the first edition serious consideration of this theory began in Chapter 12. The result of this revision is a remarkable improvement in organisation. The preface to the second edition mentions "space saved by streamlining," but it must be emphasised that the streamlining is in organisation, not in exposition. Explanations and examples have been expanded so that what was already an outstanding bit of exposition has been noticeably improved.

2.9. Review by: Kenneth O May.
American Scientist 46 (2) (1958), 150A, 152A.

The first edition of this justly famous book was so successful that the author was stimulated to prepare a second edition even before the appearance of the second volume, which one may hope will not be too long delayed. In the first edition, by limiting himself to discrete distributions, the author was able to pre sent probability theory in both an elementary and a sophisticated manner. The approach was fresh, modern, and deeper in its penetration than previous books. It could be and was read by many mature people with relatively modest mathematical preparation. It could also be read with profit and pleasure by specialists in the field of probability. The second edition has all the good features of the first. The revisions are primarily reorganizations to increase flexibility and ease of classroom use. ... Even more than the first edition, this book is to be heartily recommended to the scientific public.

2.10. Review by: Richard Cook.
Mathematics Magazine 32 (1) (1958), 43-44.

Volume I of William feller's "An Introduction to Probability Theory and its Applications" now appears in a second edition, published in September, 1957, by John Wiley & Sons. The book retains its former general plan and basic content, both of which have achieved such tremendous popularity in only seven years, but adds extensive revisions and stimulating new material that enhance the original substance. Dr Feller restricts his subject matter to discrete sample spaces and discrete variables. In this way, he allows himself greater detail in the treatment of many typical problems and explanations of the probabilistic approach to them.

2.11. Review by: H Richter.
Econometrica 27 (3) (1959), 534-536.

The rapid appearance of the second edition of this textbook demonstrates its popularity as an introduction into the calculus of probability. The main structure of the book is not changed. But more sections than in the first edition are now designated as unnecessary for a first reading. The other parts of the book form a complete treatise starting with the fundamental notions of probability and ending with Markoff processes. This part of the text is accessible to readers who have no very advanced mathematical training. The treatment is purely mathematical, but the intuitive background is stressed by the author, especially in connection with the introduction of the various axioms. He restricts himself to a discussion of enumerably infinite simple events. This procedure makes it possible to present to the reader the formal mathematical development, but also to give him the intuitive insight which is necessary for the application of the theory to practical problems. When examples are introduced, the language is also frequently intuitive, but the author demonstrates in several places also, the insufficiency of intuition. ... The reader who is familiar with measure theory might feel that the reluctance of the author to discuss anything more complicated than enumerably infinite probability fields is an unnecessary obstacle. This point is also mentioned in the text. But the particular value of the book consists in the fact, that in contradistinction to other modern texts in the field the reader is given an introduction to modern problems of the probability calculus without a lot of complicated mathematics. But the reader also feels that it will be necessary to abandon this restriction in the second volume of the treatise which has been announced. Let us hope that it will be published soon.

2.12. Review by: F N David.
Biometrika 45 (1/2) (1958), 287.

This is the second edition of a book which was first issued in 1950. There are some alterations but not many. The major difference from the first edition appears in the insertion of Chapter III, 'Fluctuations in Coin Tossing and Random Walks', where the random walk occurs a little earlier than originally. During the seven years since this book was issued it has rapidly established itself as a classic in probability theory connected with the discrete variable. One notes, hopefully, that it is Vol. I.

2.13. Review by: M T L B.
Journal of the Institute of Actuaries (1886-1994) 84 (2) (1958), 232-234.

In the Journal review of the first edition of this book we read : 'there are good grounds for believing that for many years it will be the probability textbook par excellence'. This prediction has certainly been true of the eight years since publication ; in breadth of content, clarity of exposition and wealth of illustrative applications, the book is unique. Described by the reviewer as 'impeccable' it has very nearly justified this bold epithet, for the errors that have come to light after close study are indeed few and slight. This freedom from fault, which must have contributed not a little to the success of the work, is the more remarkable when we recall that its 400-odd pages contained a very considerable amount of new material. If the book has a weakness at all, this lies in the misleadingly modest title, which must have lured many an unsuspecting student in search of an elementary textbook to explore its pages before recoiling in pained astonishment that this great work should be a mere introduction to the subject of his studies. The student who persevered, however, will have been rewarded beyond his expectations, for in the process of learning he will have embarked upon an intellectual adventure of the highest order. How, then, can any second edition be an improvement upon such a brilliantly successful original? Examination reveals that Professor Feller has accomplished the seemingly impossible task of bettering his own masterpiece by means of an increase in scope, and a more thorough concatenation of related topics, rather than by any amelioration of the existing detail. The changes have been made so deftly that the author has nowhere disturbed the continuity, or marred the beauty, of the unfolding pattern of his work; rather has his practised hand blended it anew into yet more attractive form. Those who admired the first edition - and they surely include all its readers - may be unreservedly advised to read the second; high though their hopes may run, they will assuredly be delighted.

2.14. Review by: U Grenander.
Mathematical Reviews MR0088081 (19,466a).

The first edition of this book appeared in 1950. The second edition contains the same material but rearranged and with substantial additions. Especially the third chapter on fluctuations in coin tossing and random walks contains much new and gives a thorough discussion of the arc sine law and related questions. ... Also new is the twelfth chapter on compound distributions and branching processes, which contains a fuller discussion of these topics than the first edition. ... As in the first edition the exposition is mathematically rigorous and at the same time elegant and lucid. This fascinating book will remain a standard textbook of mathematical probability for many years to come.

2.15. Review by: Editors.
Bulletin American Mathematical Society 64 (6) (1958), 393.

The first edition was reviewed in this Bulletin ... . The present edition retains the same general plan and spirit as the first, but changes appear on almost every page. Many new examples and exercises have been added, references have been brought up to date, statements and proofs clarified or simplified, and there are many rearrangements. A new chapter treating fluctuations in coin tossing and the arc sine law by elementary combinatorial methods has been added early in the book. The total length of the book has been increased by about 40 pages.