The axioms and the fundamental theorems that follow from them are presented at the beginning of the text, along with a brief history of the origins of probability theory. Chapter 2 launches into some basic mathematical techniques for counting the outcomes of events. Mathematical methods of counting go under the name of combinatorial analysis, which is also used in statistics, information theory, coding and decoding, production scheduling, and many other fields. This chapter is followed by a discussion of conditional probability, in which the outcome of the event being sampled depends on the results of some other event, and independence, in which it doesn't. Next, Ghahramani introduces the concept of random variables.
Random variables represent the possible outcomes of an event.
The properties of discrete random variables are the subject of Chapters 4 and 5. In Chapter 6, the theory is extended to continuous random variables, such as the time of a random event-in which the number of possible outcomes is infinite, and the probability of any given outcome is theoretically zero. The next few chapters delve deeper into the subject, examining special continuous distributions, and joint distributions of two or more continuous random variables considered simultaneously. Special emphasis is paid to the Poisson distribution because, as the author points out, it is the mathematical model of an enormous class of phenomena. In all the previous instances, probability functions, distribution functions or density functions are known.