Finite Markov Chains is an undergraduate introduction to probability theory and a reference for those in fields outside of mathematics. It provides examples from physics, economics, and the life sciences, and explains the central techniques of Markov chains, such as matrix operations for transition matrices. The book is suitable for an undergraduate course and includes exercises for Chapters II–VI. A new edition of a sequel, Denumerable Markov Chains, is published by Springer-Verlag as Volume 40 of the Graduate Texts in Mathematics series. The book is divided into three parts: Chapter I covers prerequisites, Chapters II–VI develop the theory of Markov chains, and Chapter VII applies the theory to various fields. The book includes appendices with formulas and examples, and a detailed table of contents. The authors acknowledge support from the National Science Foundation and the Dartmouth Mathematics Project, and thank research assistants for their contributions. The book is designed for an undergraduate mathematics course, with proofs carried out by elementary methods. It is suitable for a one-semester course on Markov chains and their applications. The book includes a new appendix on the generalization of a fundamental matrix. The authors also discuss the use of pseudo-inverses in computing basic quantities for ergodic Markov chains. The book covers topics such as absorbing and ergodic chains, first passage times, and applications to various fields. The book is written in a clear and concise manner, with a focus on practical applications and examples. The authors provide a detailed explanation of the concepts and techniques of Markov chains, making it accessible to students and researchers in various fields. The book is well-structured, with a logical flow of topics and a comprehensive coverage of the subject matter. The authors have also included a detailed table of contents and appendices, making it a valuable reference for those interested in Markov chains. The book is written in English and is suitable for both undergraduate and graduate students. The authors have also included a detailed explanation of the concepts and techniques of Markov chains, making it accessible to students and researchers in various fields. The book is well-structured, with a logical flow of topics and a comprehensive coverage of the subject matter. The authors have also included a detailed table of contents and appendices, making it a valuable reference for those interested in Markov chains. The book is written in English and is suitable for both undergraduate and graduate students.Finite Markov Chains is an undergraduate introduction to probability theory and a reference for those in fields outside of mathematics. It provides examples from physics, economics, and the life sciences, and explains the central techniques of Markov chains, such as matrix operations for transition matrices. The book is suitable for an undergraduate course and includes exercises for Chapters II–VI. A new edition of a sequel, Denumerable Markov Chains, is published by Springer-Verlag as Volume 40 of the Graduate Texts in Mathematics series. The book is divided into three parts: Chapter I covers prerequisites, Chapters II–VI develop the theory of Markov chains, and Chapter VII applies the theory to various fields. The book includes appendices with formulas and examples, and a detailed table of contents. The authors acknowledge support from the National Science Foundation and the Dartmouth Mathematics Project, and thank research assistants for their contributions. The book is designed for an undergraduate mathematics course, with proofs carried out by elementary methods. It is suitable for a one-semester course on Markov chains and their applications. The book includes a new appendix on the generalization of a fundamental matrix. The authors also discuss the use of pseudo-inverses in computing basic quantities for ergodic Markov chains. The book covers topics such as absorbing and ergodic chains, first passage times, and applications to various fields. The book is written in a clear and concise manner, with a focus on practical applications and examples. The authors provide a detailed explanation of the concepts and techniques of Markov chains, making it accessible to students and researchers in various fields. The book is well-structured, with a logical flow of topics and a comprehensive coverage of the subject matter. The authors have also included a detailed table of contents and appendices, making it a valuable reference for those interested in Markov chains. The book is written in English and is suitable for both undergraduate and graduate students. The authors have also included a detailed explanation of the concepts and techniques of Markov chains, making it accessible to students and researchers in various fields. The book is well-structured, with a logical flow of topics and a comprehensive coverage of the subject matter. The authors have also included a detailed table of contents and appendices, making it a valuable reference for those interested in Markov chains. The book is written in English and is suitable for both undergraduate and graduate students.