Probability Theory

How can randomness and uncertainty be described with mathematical precision? And why do many random processes lead to remarkably stable patterns and laws?This textbook provides a systematic, mathematically rigorous, and accessible introduction to the fundamental concepts of probability theory. It is aimed primarily at mathematics students from the third semester onward, as well as students in related fields.The book covers random experiments, probability measures, random variables, important probability models, conditional probability, Bayes' theorem, independence, and the concepts of expectation, variance, and covariance. Two fundamental limit theorems, the Law of Large Numbers and the Central Limit Theorem, are also developed and explained.Alongside classical examples, the book presents numerous applications from computer science and machine learning. A wide range of exercises, accompanied by detailed and carefully structured solutions, helps readers apply the mathematical concepts confidently and develop a deeper understanding of the underlying theory.A rigorous introduction to probability theory for university-level study, combining mathematical precision, practical relevance, and clear explanations.