The aim of writing this book is to systematically present the matter so as to bring maximum clarity. Dealing with probabilities of complex events one must have a back ground of set theory and permutations and combinations. The author has taken special efforts to carefully present the different theories; and is confident that the book will clear the common doubts and confusions in student's mind regarding probability of complex events, set theory, permutations and combinations. Several illustrative solved examples in each chapter help students to assimilate the theories. A separate chapter has a set of twenty examples that covers the entire theory covered in earlier chapters. The last chapter gives applications from different fields. Efforts were taken to study different applications to arrive at a general method involving four steps. It is hoped; this book is useful as a reference book to higher secondary, graduate and post graduate students of different faculties. People who areinterested in work on random processes, predictions of irregular, uncontrollable events, may find the book interesting and useful.