This introductory chapter lays the groundwork for understanding probability, a fundamental concept in the real world. We start by exploring different interpretations of probability, such as classical, empirical, and subjective, providing insights into how we perceive uncertainty. To speak the language of probability, we delve into set theory. This includes representing sets in the roster and set builder forms, understanding common sets, and exploring the notion of membership, subsets, supersets, and their properties. We also introduce cardinality, special sets, and various set operations like union, intersection, and complement. Constructing the probability space is a key step. We define the sample space and event space and introduce the probability measure. The chapter proceeds by setting the rules with the Axioms of Probability, encompassing nonnegativity, normalization, and additivity. These axioms form the foundation upon which the different probabilities are based. Corollaries derived from the axioms, such as the complement rule, upper bound of probability, inclusion-exclusion principle, and the multiplication rule for independent events, expand our understanding of probability. Conditional probability is explored, allowing us to assess the probability of events given specific conditions. Finally, we discuss Bayes' theorem, a powerful tool for updating beliefs based on new evidence. We discuss its significance, the base rate fallacy, and how it addresses the need to consider prior probabilities when making probabilistic inferences.
In essence, this chapter sets the stage by laying the groundwork and introducing the core concepts necessary for a deeper understanding. In the next chapter, we'll extend our knowledge by introducing random variables, which provide a quantitative framework for understanding uncertain outcomes. Additionally, probability distributions will enable us to model real-world scenarios with precision and analyze the associated probabilities.