The theoretical foundations of probability have been developed over nearly four centuries. This legacy is both a blessing and a curse. It is a blessing because it is so rich and diverse, offering many results whose relations to one another, obscure at first glance, form a beautiful logical tapestry. It is a curse because appreciating this structure requires many, often complex, proofs, and textbooks tend to address this complexity in one of two ways. Some offer an informal summary that leaves students with knowledge of what probability has to tell them (often as background for an applied curriculum in statistics) but without knowledge of why these principles are so. Other more formal presentations present a litany of definition, lemma, and theorem that can bewilder the newcomer.

Miller’s volume may be the friendliest example of the formal variety of text currently available. It unapologetically takes the student through proof after proof, but in a way that anticipates some of the stumbling blocks encountered in more conventional texts. For example:

• Conventional proofs are often terse. Miller is generous with paper and ink, yielding a massive volume (700 7-inch x 10-inch pages).
• Texts usually present one proof for each theorem, and offer no help to a reader who has lost the thread. Miller provides multiple proofs for the most important and complex theorems, approaching them from different directions to help the student gain multiple perspectives on what is going on.
• The goal of mathematics education ought to be not just the understanding of how existing results are proved but teaching the student to develop new proofs. Miller devotes considerable attention to explaining the intuitions that lie behind the proofs he presents and includes an appendix that explains and illustrates 11 different proof procedures.
• Formal texts often skimp on illustrative material, but Miller draws freely on informal examples, including basketball, sports betting, and casino games.
• Proofs and simulations are often seen as competing approaches to mathematics, but Miller unites the two, providing numerous computer programs to allow the reader to test the results of theorems with computational experiments.

The book begins by motivating the subject with the ubiquitous birthday problem, estimating the odds on a basketball shootout, and wagering on the Super Bowl. After a chapter on foundations (including set theory, probability spaces, and axioms), Miller uses a series of card games to introduce principles of counting. A chapter on conditional probability is followed by two more on counting.

This theoretical introduction is followed by five chapters on random variables, five on specific distributions, and five on limit theorems, with multiple proofs of the central limit theorem. The last four chapters treat additional topics, including hypothesis testing (for students heading toward statistics), Markov processes, least squares, and discussion of the marriage problem and the Monty Hall problem. Appendices discuss proof techniques, some fundamental results from analysis, further details on countability in infinite sets that underlie some of the earlier results, and further information on the central limit theorem from the perspective of complex analysis.

The breadth of the book’s coverage and its clear, informal tone in addressing highly formal problems remind one of a friendly professor offering unlimited office hours, and the book will be a highly accessible supplement for students working through another, more conventional text. But it includes a rich array of exercises and would serve as an excellent primary text at the upper undergraduate level. One could wish that the computational experiments, presented in Mathematica, were offered for a less expensive platform such as Python, but in a university environment, most students will have access to student-priced versions of Mathematica. This detail is a minor quibble for a volume that deserves to be widely known in educational circles and will likely find its way to the shelves of practicing statisticians who wish to probe below the surface of fundamental theorems that they have learned by rote.

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