We discuss the concept of a ‘random event’. The classical and statistical approaches used to formalize the notion of probability are described, along with the basic concepts of set theory and measure theory. The Kolmogorov approach for axiomatizing probability theory is presented. The probability space is introduced. The axioms of probability theory are presented, together with the addition and multiplication theorems. The notion of a scalar random variable is formalized. We present ways to describe a random variable in terms of the distribution function, probability density function, and moments, including in particular, the expectation and variance. Examples of scalar random variables with different distribution laws are presented. Methods for describing a scalar random variable are generalized to a vector random variable. The transformation of random variables and arithmetic operations on them are briefly examined.