Ghahramani’s book is intended as a text for undergraduate courses in probability, though it could also be used at the graduate level. It is especially geared towards probability classes designed to prepare students for the actuarial exams, though it is general enough to be useful in non-specific courses as well. It largely takes a historically motivated and axiomatic approach to probability. The book is written at an extremely high level that might not make it suitable for all undergraduate probability courses.

The book is well-written and covers a full range of topics in probability while providing many examples across several disciplines and it includes many exercises for students to attempt, separated into two different levels of difficulty. There are, however, several sections with no or very few exercises. It also provides self-quizzes and tests after nearly every section and chapter (though it is odd that the author assigns possible points for each question on a self-quiz a student would give themselves). The book also has a companion website with additional content as well as a solutions manual and test bank available for instructors.

The book is written at an extremely high level for an undergraduate text and assumes a great deal of knowledge is already possessed by the students. This may not be appropriate for some schools, depending on the order in which students typically take their upper division classes. Specifically, it assumes knowledge of set theory, the theory of analysis, measure theory, computing theory, linear algebra, and multivariable calculus.

The book is strongly text based, with very few accompanying pictures, diagrams, or tables that would help a student comprehend various examples. Many definitions and examples are also placed in the middle of large paragraphs and mixed together, which makes it hard to understand what is being presented. Even mathematical equations are given inline instead of centered on their own line, which makes them hard to read. It also tends to dive directly into definitions and theory without providing any leading explanation or examples. And the examples that follow the theory often are presented with the more complex ones coming first before the more simple and basic ones. These problems are less important if the level of the students is sufficiently advanced, especially as it is very thorough in its inclusion of topics, but more typical undergraduates often require more lead up to complex theory and an easier to read and follow text.

The presentation of applications is thorough and excellent. In addition to the previously mentioned actuarial applications, there are several other applications including an exploration of genetics. I particularly appreciated that the author took pains to explain how DNA matches can be misunderstood in the courtroom, and how this applies to the Prosecutor’s Fallacy.

I caution users of the book that there are a large number of examples and exercises that assume gender binarism. While I understand the ease of using such examples when seeking binary choices, my own school has asked me to remove all wording from my classes that implies a strictly binary nature of gender and I assume they are not alone in asking this.

In summary, the book was a particularly good and thorough read, but it felt like reading a graduate text instead of one pitched towards undergraduates. It covered all aspects of the fundamentals of probability and provided many examples (though often in an odd order, with the simpler examples often appearing after more complex ones). It would be a good text for a graduate-level probability course and could be used in an undergraduate course if the students have already taken multivariable calculus, linear algebra, real analysis, and some type of course covering set theory in depth.

About the Reviewer: Meghan De Witt (

mdewitt@stac.edu) is a professor at St. Thomas Aquinas College whose primary interests are unusual applications of group theory, number theory, and mathematical outreach programs.