Quantitative Models Play a Crucial

in population genetics. Mathematical analysis has shown how different evolutionary processes interact, and statistical methods have made it possible to interpret observations and experiments. The availability of computing power and the flood of genomic data have made quantitative methods even more prominent in recent years. Although we largely avoid explicit mathematics in Evolution, mathematical arguments lie behind much of what is discussed. In this chapter, the basic mathematical methods that are used in evolutionary biology are outlined. The chapters in the book can be understood in a qualitative way without this chapter, and that may be appropriate for an introductory course. Similarly, many of the problems (online) associated with Chapters 13–23 involve only elementary algebra and do not require the mathematical methods explained in this chapter. However, reading through this chapter and working through the problems that depend on it will give you a much stronger grasp of evolutionary modeling and will allow you to fully engage with the most recent research in the field. Most students will have had some basic courses in statistics, focusing on methods for testing alternative hypotheses and making estimates from the data. Although we do emphasize the evidence that is the basis for our understanding of evolutionary biology, we do not do more than touch on methods for statistical inference. However, our understanding of the evolutionary processes does fundamentally rest on knowledge of probability: Indeed, much of modern probability theory has been motivated by evolutionary problems (see the Random Processes section of this chapter). This is true both when we follow the proportions of different genotypes within a population and, more obviously, when we follow random evolutionary processes (Chapter 15) in explicitly probabilistic models. (The importance of probability will become apparent when you work through the problems [online].) In this chapter, we discuss the definitions and meaning of probability and summarize the basic theory of probability. This chapter begins with an explanation of how mathematical theory can be used to model reproducing populations. Next, we discuss how deterministic processes (i.e., those with no random element) can be modeled by following genotype frequencies. Then, random processes are examined, summarizing important probability distribua/a