This book provides an introduction to elementary probability and some of its applications. The word elementary means that the book does not need the abstract Lebesgue measure and integration theory, only elementary calculus is used. The strong feature of the textbook is a choice of good examples. Each theoretical explanation is accompanied by a large number of examples and followed by worked examples incorporating a cluster of exercises. The examples and exercises illustrate the treated topics and help the student solve the kind of problems typical of examinations. Each chapter concludes with problems. Solutions to many of these appear in an appendix, together with solutions to most of exercises. The second edition of the book contains some new sections. A new section provides a first introduction to elementary properties of martingales, which is now occupying a central position in modern probability. Another section provides an elementary introduction to Brownian motion, diffusion, and the Wiener process, which has underpinned much of classical financial mathematics, such as the Black-Scholes formula for pricing options. Optional stopping and its applications are introduced in the context of these important stochastic models, together with several associated new examples and exercises. The list of chapters: Introduction, Probability, Conditional Probability and Independence, Counting, Random Variables: Distribution and Expectation, Random Vectors: Independence and Dependence, Generating Functions and Their Applications, Continuous Random Variables, Jointly Continuous Random Variables, and Markov Chains. Some sections can be omitted at a first reading (e.g. Generating Functions, Markov Chains). The book is suitable for a first university course in probability and very useful for self-study.