In this book, we present the rigorous foundation of probability, within the framework of a second, modern look at the subject. We cover the concepts chapter by chapter: measurable space, measures and probabilities, random variables, distribution function, Kolmogorov's 0-1 law and Borel-Cantelli lemmas, integration and mean value, convergence of sequences of random variables and laws of large numbers, weak convergence and characteristic function (including the central limit theorem). In terms of structure, each chapter is divided into two main sections: the first section contains the necessary presentation of the theory, with detailed comments and observations on the theorems and solved examples. The second section contains a multitude of solved sample problems, approximately 450 in total throughout the book and of various difficulty levels, some of which have been exam topics in mathematical, engineering, and economic schools.