Advanced Probabilities:Conforming to the first-year Master of Computer Science – Quantum Computing Curriculum (2024/2025)

Abstract

This booklet has been designed for first-year Master’s students in Computer Science, option: Quantum Computing. By blending theoretical foundations with illustrative examples, it aims to strengthen both intuition and analytical rigor. The material not only introduces students to key probabilistic tools but also prepares them to apply these concepts directly within the framework of quantum information theory, quantum algorithms, and related computational paradigms. This booklet is intended as a concise yet comprehensive introduction to probability and random variables. Each section is accompanied by illustrative examples, and at the end of each chapter, students are encouraged to work on additional exercises to reinforce theoretical concepts and enhance problem-solving skills. The booklet is organized into four main chapters. The first chapter introduces the fun- damental principles of probability, including counting techniques, sample spaces, and the concept of independence. Conditional probability and Bayes’ theorem are presented early on to provide a solid foundation for probabilistic reasoning. Building on these basics, the second chapter explores discrete and continuous random variables, covering key concepts such as probability distributions, expected value, variance, and standard deviation. Special attention is given to widely used distributions, including the Bernoulli, Binomial, Poisson, Normal, Exponential, and Gamma distributions, which play a central role in statistics, data science, and various applied fields. The third chapter addresses probability distributions of combined random variables. Topics such as joint distributions, conditional expectations, and their applications are detailed to enable learners to analyze more complex systems involving multiple sources of randomness. Finally, the last chapter focuses on conditional probabilities and independence, empha- sizing the role of conditional distributions, their properties, and their importance in both theoretical and applied contexts