The book is an initial course on probability theory and its applications, for the understanding of which knowledge of mathematics is necessary in some reasonable, but not necessarily wide boundaries. It brings the reader to a level that can be called a good average level. The fundamental novelty of the present exposition is that the whole theory is based on the axiomatization of the concept of mathematical expectation, and not the concept of a probabilistic measure. The authors believe that this approach has great advantages even when presenting probability theory at the introductory level.