7. Introduction to probability

This chapter serves as a foundational exploration of probability, bridging the gap between descriptive statistics and the deeper domain of inferential statistics. Beginning with a reflection on the interplay between probability and statistical inference, it emphasises the importance of understanding probability theory as the backbone of statistical reasoning. Through examples such as polling data and the representativeness of samples, the chapter demonstrates how probability allows us to make informed inferences about populations from limited data. It differentiates probability from statistics, defining probability as the study of chance events under known conditions, while statistics seeks to infer unknown truths from observed data. By contrasting the two, the chapter prepares readers to appreciate probability as a tool for robust scientific inquiry.

The chapter delves into the philosophical underpinnings of probability, introducing the frequentist and Bayesian perspectives. While the frequentist approach views probability as a long-run frequency of events, the Bayesian perspective interprets probability as a degree of belief, contextualised by prior knowledge. Both viewpoints are explored with examples to illustrate their practical implications and limitations. Additionally, the chapter introduces key probability distributions, including the binomial and normal distributions, highlighting their relevance in statistical applications. By covering the fundamental rules of probability and addressing nuances like probability density, the chapter equips readers with the conceptual clarity needed for subsequent discussions on statistical inference.