3. Probability Theory

Statistics is a science that is concerned with principles, methods, and techniques for collecting, processing, analyzing, presenting, and interpreting (numerical) data. Statistics can be divided roughly into descriptive statistics (Chap. 1) and inferential statistics (Chap. 2), as we have already suggested. Descriptive statistics summarizes and visualizes the observed data. It is usually not very difficult, but it forms an essential part of reporting (scientific) results. Inferential statistics tries to draw conclusions from the data that would hold true for part or the whole of the population from which the data is collected. The theory of probability, which is the topic of the next two theoretical chapters, makes it possible to connect the two disciplines of descriptive and inferential statistics. We have already encountered some ideas from probability theory in the previous chapter. To start with, we discussed the probability of selecting a specific sample \(\pi _k\) and we briefly defined the notion of probability based on the throwing of a dice. In this chapter we work out these ideas more formally and discuss the probabilities of events; we define probabilities and discuss how to calculate with probabilities. In the previous chapter, when discussing bias, we have also encountered the expected population parameter \(\mathbb {E}(T)\), but we have not yet detailed what expectations are exactly; this is something we cover in Chap. 4.