The book Elements of Probability Theory and Its Applications is a text designed primarily for undergraduate beginners who intend to major in statistics and also for their instructors. It can equally be used by the readers of physical and social sciences and other disciplines having some background in mathematics and calculus. It provides an introductory mathematical treatment to probability theory and statistics. However, the level of mathematical treatment is not that rigorous in most parts of the book except for the last two chapters, which advanced-level students, instructors, and researchers will find useful. This book can also be used as a reference text as many of the results, theorems, and rules discussed have applications in diverse areas.This book focuses on the evolution of probability theory with various concepts, rules, and laws of probability; concepts of random variables and their probability distributions; distributions of functions of random variables; tools such as mathematical expectation, generating functions, sets, and combinatorial methods for analysing the behaviour and properties of probability distributions and for solving probability problems; and a thorough discussion on important discrete and continuous probability distributions. These topics are presented in a logical order in the first nine chapters based on which one can design one semester or two-quarters of an under-graduate course on probability theory. The instructors may skip some of the mathematical details and topics while offering a one-semester course to non-major students.