Probability And Statistics With Reliability Queuing And Computer Science Applications

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Probability and Statistics with Reliability, Queuing, and Computer Science Applications

Author : Kishor S. Trivedi
Publisher : John Wiley & Sons
ISBN 13 : 1119314208
Total Pages : 880 pages
Book Rating : 4.1/5 (193 download)

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Book Synopsis Probability and Statistics with Reliability, Queuing, and Computer Science Applications by : Kishor S. Trivedi

Download or read book Probability and Statistics with Reliability, Queuing, and Computer Science Applications written by Kishor S. Trivedi and published by John Wiley & Sons. This book was released on 2016-06-30 with total page 880 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible introduction to probability, stochastic processes, and statistics for computer science and engineering applications Second edition now also available in Paperback. This updated and revised edition of the popular classic first edition relates fundamental concepts in probability and statistics to the computer sciences and engineering. The author uses Markov chains and other statistical tools to illustrate processes in reliability of computer systems and networks, fault tolerance, and performance. This edition features an entirely new section on stochastic Petri nets—as well as new sections on system availability modeling, wireless system modeling, numerical solution techniques for Markov chains, and software reliability modeling, among other subjects. Extensive revisions take new developments in solution techniques and applications into account and bring this work totally up to date. It includes more than 200 worked examples and self-study exercises for each section. Probability and Statistics with Reliability, Queuing and Computer Science Applications, Second Edition offers a comprehensive introduction to probability, stochastic processes, and statistics for students of computer science, electrical and computer engineering, and applied mathematics. Its wealth of practical examples and up-to-date information makes it an excellent resource for practitioners as well. An Instructor's Manual presenting detailed solutions to all the problems in the book is available from the Wiley editorial department.

Probability and Statistics with Reliability, Queuing, and Computer Science Applications

Author : Kishor S. Trivedi
Publisher : John Wiley & Sons
ISBN 13 : 0471333417
Total Pages : 851 pages
Book Rating : 4.4/5 (713 download)

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Book Synopsis Probability and Statistics with Reliability, Queuing, and Computer Science Applications by : Kishor S. Trivedi

Download or read book Probability and Statistics with Reliability, Queuing, and Computer Science Applications written by Kishor S. Trivedi and published by John Wiley & Sons. This book was released on 2001-11-28 with total page 851 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible introduction to probability, stochastic processes, and statistics for computer science and engineering applications Second edition now also available in Paperback. This updated and revised edition of the popular classic first edition relates fundamental concepts in probability and statistics to the computer sciences and engineering. The author uses Markov chains and other statistical tools to illustrate processes in reliability of computer systems and networks, fault tolerance, and performance. This edition features an entirely new section on stochastic Petri nets—as well as new sections on system availability modeling, wireless system modeling, numerical solution techniques for Markov chains, and software reliability modeling, among other subjects. Extensive revisions take new developments in solution techniques and applications into account and bring this work totally up to date. It includes more than 200 worked examples and self-study exercises for each section. Probability and Statistics with Reliability, Queuing and Computer Science Applications, Second Edition offers a comprehensive introduction to probability, stochastic processes, and statistics for students of computer science, electrical and computer engineering, and applied mathematics. Its wealth of practical examples and up-to-date information makes it an excellent resource for practitioners as well. An Instructor's Manual presenting detailed solutions to all the problems in the book is available from the Wiley editorial department.

PROBABILITY AND STATISTICS WITH RELIABILITY, QUEUING, AND COMPUTER SCIENCE APPLICATIONS

Author : KISHOR SHRIDHARBHAI TRIVEDI
Publisher : PHI Learning Pvt. Ltd.
ISBN 13 : 9788120305083
Total Pages : 646 pages
Book Rating : 4.3/5 (5 download)

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Book Synopsis PROBABILITY AND STATISTICS WITH RELIABILITY, QUEUING, AND COMPUTER SCIENCE APPLICATIONS by : KISHOR SHRIDHARBHAI TRIVEDI

Download or read book PROBABILITY AND STATISTICS WITH RELIABILITY, QUEUING, AND COMPUTER SCIENCE APPLICATIONS written by KISHOR SHRIDHARBHAI TRIVEDI and published by PHI Learning Pvt. Ltd.. This book was released on 1988-01-01 with total page 646 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to probability, stochastic processes, and statistics for students of computer science, electrical/computer engineering, reliability engineering and applied mathematics. It prepares the student for solving practical stochastic modelling problems, and for the more advanced courses on queuing or reliability theory. The text emphasizes on applications, illustrating each theoretical concept by solved examples relating to algorithm analysis or communication related problems.The prerequisites are a knowledge of calculus, a course on introduction to computer programming, and an understanding of computer organization. The book is also suitable for self-study by computer professionals and mathematicians interested in applications.

Probability and Statistics for Computer Scientists

Author : Michael Baron
Publisher : CRC Press
ISBN 13 : 1498760600
Total Pages : 475 pages
Book Rating : 4.4/5 (987 download)

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Book Synopsis Probability and Statistics for Computer Scientists by : Michael Baron

Download or read book Probability and Statistics for Computer Scientists written by Michael Baron and published by CRC Press. This book was released on 2013-08-05 with total page 475 pages. Available in PDF, EPUB and Kindle. Book excerpt: Student-Friendly Coverage of Probability, Statistical Methods, Simulation, and Modeling ToolsIncorporating feedback from instructors and researchers who used the previous edition, Probability and Statistics for Computer Scientists, Second Edition helps students understand general methods of stochastic modeling, simulation, and data analysis; make o

Reliability and Availability Engineering

Author : Kishor S. Trivedi
Publisher : Cambridge University Press
ISBN 13 : 1107099501
Total Pages : 729 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Reliability and Availability Engineering by : Kishor S. Trivedi

Download or read book Reliability and Availability Engineering written by Kishor S. Trivedi and published by Cambridge University Press. This book was released on 2017-08-03 with total page 729 pages. Available in PDF, EPUB and Kindle. Book excerpt: Learn about the techniques used for evaluating the reliability and availability of engineered systems with this comprehensive guide.

Queueing Networks and Markov Chains

Author : Gunter Bolch
Publisher : John Wiley & Sons
ISBN 13 : 0471565253
Total Pages : 901 pages
Book Rating : 4.4/5 (715 download)

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Book Synopsis Queueing Networks and Markov Chains by : Gunter Bolch

Download or read book Queueing Networks and Markov Chains written by Gunter Bolch and published by John Wiley & Sons. This book was released on 2006-04-14 with total page 901 pages. Available in PDF, EPUB and Kindle. Book excerpt: Critically acclaimed text for computer performance analysis--now in its second edition The Second Edition of this now-classic text provides a current and thorough treatment of queueing systems, queueing networks, continuous and discrete-time Markov chains, and simulation. Thoroughly updated with new content, as well as new problems and worked examples, the text offers readers both the theory and practical guidance needed to conduct performance and reliability evaluations of computer, communication, and manufacturing systems. Starting with basic probability theory, the text sets the foundation for the more complicated topics of queueing networks and Markov chains, using applications and examples to illustrate key points. Designed to engage the reader and build practical performance analysis skills, the text features a wealth of problems that mirror actual industry challenges. New features of the Second Edition include: * Chapter examining simulation methods and applications * Performance analysis applications for wireless, Internet, J2EE, and Kanban systems * Latest material on non-Markovian and fluid stochastic Petri nets, as well as solution techniques for Markov regenerative processes * Updated discussions of new and popular performance analysis tools, including ns-2 and OPNET * New and current real-world examples, including DiffServ routers in the Internet and cellular mobile networks With the rapidly growing complexity of computer and communication systems, the need for this text, which expertly mixes theory and practice, is tremendous. Graduate and advanced undergraduate students in computer science will find the extensive use of examples and problems to be vital in mastering both the basics and the fine points of the field, while industry professionals will find the text essential for developing systems that comply with industry standards and regulations.

Introduction to Probability and Statistics for Science, Engineering, and Finance

Author : Walter A. Rosenkrantz
Publisher : CRC Press
ISBN 13 : 158488813X
Total Pages : 680 pages
Book Rating : 4.5/5 (848 download)

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Book Synopsis Introduction to Probability and Statistics for Science, Engineering, and Finance by : Walter A. Rosenkrantz

Download or read book Introduction to Probability and Statistics for Science, Engineering, and Finance written by Walter A. Rosenkrantz and published by CRC Press. This book was released on 2008-07-10 with total page 680 pages. Available in PDF, EPUB and Kindle. Book excerpt: Integrating interesting and widely used concepts of financial engineering into traditional statistics courses, Introduction to Probability and Statistics for Science, Engineering, and Finance illustrates the role and scope of statistics and probability in various fields. The text first introduces the basics needed to understand and create

Probability & Statistics With Reliability, Queuing And Computer Science Applications, 2Nd Ed

Author : Kishor S. Trivedi
Publisher : John Wiley & Sons
ISBN 13 : 9788126518531
Total Pages : 852 pages
Book Rating : 4.5/5 (185 download)

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Book Synopsis Probability & Statistics With Reliability, Queuing And Computer Science Applications, 2Nd Ed by : Kishor S. Trivedi

Download or read book Probability & Statistics With Reliability, Queuing And Computer Science Applications, 2Nd Ed written by Kishor S. Trivedi and published by John Wiley & Sons. This book was released on 2008-10-15 with total page 852 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is important to our developing list of computer science titles. Trivedi's book is a true classic and will be well received in the market. The subject lies at the core of many applications in computer science, signal processing, and communications. · Introduction· Discrete Random Variables· Continuous Random Variables· Expectation· Conditional Distribution and Expectation· Stochastic Processes· Discrete-Time Markov Chains· Continuous-Time Markov Chains· Networks of Queues· Statistical Inference · Regression and Analysis of Variance

Statistics and Probability for Engineering Applications

Author : William DeCoursey
Publisher : Elsevier
ISBN 13 : 0080489753
Total Pages : 396 pages
Book Rating : 4.0/5 (84 download)

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Book Synopsis Statistics and Probability for Engineering Applications by : William DeCoursey

Download or read book Statistics and Probability for Engineering Applications written by William DeCoursey and published by Elsevier. This book was released on 2003-05-14 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: Statistics and Probability for Engineering Applications provides a complete discussion of all the major topics typically covered in a college engineering statistics course. This textbook minimizes the derivations and mathematical theory, focusing instead on the information and techniques most needed and used in engineering applications. It is filled with practical techniques directly applicable on the job. Written by an experienced industry engineer and statistics professor, this book makes learning statistical methods easier for today's student. This book can be read sequentially like a normal textbook, but it is designed to be used as a handbook, pointing the reader to the topics and sections pertinent to a particular type of statistical problem. Each new concept is clearly and briefly described, whenever possible by relating it to previous topics. Then the student is given carefully chosen examples to deepen understanding of the basic ideas and how they are applied in engineering. The examples and case studies are taken from real-world engineering problems and use real data. A number of practice problems are provided for each section, with answers in the back for selected problems. This book will appeal to engineers in the entire engineering spectrum (electronics/electrical, mechanical, chemical, and civil engineering); engineering students and students taking computer science/computer engineering graduate courses; scientists needing to use applied statistical methods; and engineering technicians and technologists. * Filled with practical techniques directly applicable on the job * Contains hundreds of solved problems and case studies, using real data sets * Avoids unnecessary theory

Probability with R

Author : Jane M. Horgan
Publisher : John Wiley & Sons
ISBN 13 : 1119536987
Total Pages : 522 pages
Book Rating : 4.1/5 (195 download)

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Book Synopsis Probability with R by : Jane M. Horgan

Download or read book Probability with R written by Jane M. Horgan and published by John Wiley & Sons. This book was released on 2019-12-18 with total page 522 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides a comprehensive introduction to probability with an emphasis on computing-related applications This self-contained new and extended edition outlines a first course in probability applied to computer-related disciplines. As in the first edition, experimentation and simulation are favoured over mathematical proofs. The freely down-loadable statistical programming language R is used throughout the text, not only as a tool for calculation and data analysis, but also to illustrate concepts of probability and to simulate distributions. The examples in Probability with R: An Introduction with Computer Science Applications, Second Edition cover a wide range of computer science applications, including: testing program performance; measuring response time and CPU time; estimating the reliability of components and systems; evaluating algorithms and queuing systems. Chapters cover: The R language; summarizing statistical data; graphical displays; the fundamentals of probability; reliability; discrete and continuous distributions; and more. This second edition includes: improved R code throughout the text, as well as new procedures, packages and interfaces; updated and additional examples, exercises and projects covering recent developments of computing; an introduction to bivariate discrete distributions together with the R functions used to handle large matrices of conditional probabilities, which are often needed in machine translation; an introduction to linear regression with particular emphasis on its application to machine learning using testing and training data; a new section on spam filtering using Bayes theorem to develop the filters; an extended range of Poisson applications such as network failures, website hits, virus attacks and accessing the cloud; use of new allocation functions in R to deal with hash table collision, server overload and the general allocation problem. The book is supplemented with a Wiley Book Companion Site featuring data and solutions to exercises within the book. Primarily addressed to students of computer science and related areas, Probability with R: An Introduction with Computer Science Applications, Second Edition is also an excellent text for students of engineering and the general sciences. Computing professionals who need to understand the relevance of probability in their areas of practice will find it useful.

Systems Modeling: Methodologies and Tools

Author : Antonio Puliafito
Publisher : Springer
ISBN 13 : 3319923781
Total Pages : 315 pages
Book Rating : 4.3/5 (199 download)

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Book Synopsis Systems Modeling: Methodologies and Tools by : Antonio Puliafito

Download or read book Systems Modeling: Methodologies and Tools written by Antonio Puliafito and published by Springer. This book was released on 2018-10-16 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers ideas, methods, algorithms, and tools for the in-depth study of the performance and reliability of dependable fault-tolerant systems. The chapters identify the current challenges that designers and practitioners must confront to ensure the reliability, availability, and performance of systems, with special focus on their dynamic behaviors and dependencies. Topics include network calculus, workload and scheduling; simulation, sensitivity analysis and applications; queuing networks analysis; clouds, federations and big data; and tools. This collection of recent research exposes system researchers, performance analysts, and practitioners to a spectrum of issues so that they can address these challenges in their work.

Probability, Markov Chains, Queues, and Simulation

Author : William J. Stewart
Publisher : Princeton University Press
ISBN 13 : 1400832810
Total Pages : 777 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis Probability, Markov Chains, Queues, and Simulation by : William J. Stewart

Download or read book Probability, Markov Chains, Queues, and Simulation written by William J. Stewart and published by Princeton University Press. This book was released on 2009-07-06 with total page 777 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probability, Markov Chains, Queues, and Simulation provides a modern and authoritative treatment of the mathematical processes that underlie performance modeling. The detailed explanations of mathematical derivations and numerous illustrative examples make this textbook readily accessible to graduate and advanced undergraduate students taking courses in which stochastic processes play a fundamental role. The textbook is relevant to a wide variety of fields, including computer science, engineering, operations research, statistics, and mathematics. The textbook looks at the fundamentals of probability theory, from the basic concepts of set-based probability, through probability distributions, to bounds, limit theorems, and the laws of large numbers. Discrete and continuous-time Markov chains are analyzed from a theoretical and computational point of view. Topics include the Chapman-Kolmogorov equations; irreducibility; the potential, fundamental, and reachability matrices; random walk problems; reversibility; renewal processes; and the numerical computation of stationary and transient distributions. The M/M/1 queue and its extensions to more general birth-death processes are analyzed in detail, as are queues with phase-type arrival and service processes. The M/G/1 and G/M/1 queues are solved using embedded Markov chains; the busy period, residual service time, and priority scheduling are treated. Open and closed queueing networks are analyzed. The final part of the book addresses the mathematical basis of simulation. Each chapter of the textbook concludes with an extensive set of exercises. An instructor's solution manual, in which all exercises are completely worked out, is also available (to professors only). Numerous examples illuminate the mathematical theories Carefully detailed explanations of mathematical derivations guarantee a valuable pedagogical approach Each chapter concludes with an extensive set of exercises

Probabilistic Modelling

Author : I. Mitrani
Publisher : Cambridge University Press
ISBN 13 : 9780521585309
Total Pages : 226 pages
Book Rating : 4.5/5 (853 download)

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Book Synopsis Probabilistic Modelling by : I. Mitrani

Download or read book Probabilistic Modelling written by I. Mitrani and published by Cambridge University Press. This book was released on 1998 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probabilistic modelling is the most cost-effective means of performance and reliability evaluation of complex dynamic systems. This self-contained text will be welcomed by students and teachers for its no-nonsense treatment of the basic results and examples of their application. The only mathematical background that is assumed is basic calculus. The necessary fundamentals of probability theory are included, as well as an introduction to renewal, Poisson and Markov processes. Models arising in the fields of manufacturing, computing and communications, involving single or multiple service stations and one or more customer classes, are examined in some detail. Both exact and approximate solution methods are discussed, including recent techniques such as spectral expansion. Special attention is devoted to models of systems subject to breakdowns and repairs. Throughout the book, strong emphasis is placed on explaining the ideas behind the results and helping the reader to use them, making the book ideal for students in computer science, engineering or operations research taking courses in modern system design.

Performance and Reliability Analysis of Computer Systems

Author : Robin A. Sahner
Publisher : Springer
ISBN 13 : 9781461360056
Total Pages : 0 pages
Book Rating : 4.3/5 (6 download)

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Book Synopsis Performance and Reliability Analysis of Computer Systems by : Robin A. Sahner

Download or read book Performance and Reliability Analysis of Computer Systems written by Robin A. Sahner and published by Springer. This book was released on 2012-10-02 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Performance and Reliability Analysis of Computer Systems: An Example-Based Approach Using the SHARPE Software Package provides a variety of probabilistic, discrete-state models used to assess the reliability and performance of computer and communication systems. The models included are combinatorial reliability models (reliability block diagrams, fault trees and reliability graphs), directed, acyclic task precedence graphs, Markov and semi-Markov models (including Markov reward models), product-form queueing networks and generalized stochastic Petri nets. A practical approach to system modeling is followed; all of the examples described are solved and analyzed using the SHARPE tool. In structuring the book, the authors have been careful to provide the reader with a methodological approach to analytical modeling techniques. These techniques are not seen as alternatives but rather as an integral part of a single process of assessment which, by hierarchically combining results from different kinds of models, makes it possible to use state-space methods for those parts of a system that require them and non-state-space methods for the more well-behaved parts of the system. The SHARPE (Symbolic Hierarchical Automated Reliability and Performance Evaluator) package is the `toolchest' that allows the authors to specify stochastic models easily and solve them quickly, adopting model hierarchies and very efficient solution techniques. All the models described in the book are specified and solved using the SHARPE language; its syntax is described and the source code of almost all the examples discussed is provided. Audience: Suitable for use in advanced level courses covering reliability and performance of computer and communications systems and by researchers and practicing engineers whose work involves modeling of system performance and reliability.

An Introduction to Queueing Theory

Author : U. Narayan Bhat
Publisher : Birkhäuser
ISBN 13 : 0817684212
Total Pages : 339 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis An Introduction to Queueing Theory by : U. Narayan Bhat

Download or read book An Introduction to Queueing Theory written by U. Narayan Bhat and published by Birkhäuser. This book was released on 2015-07-09 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introductory textbook is designed for a one-semester course on queueing theory that does not require a course on stochastic processes as a prerequisite. By integrating the necessary background on stochastic processes with the analysis of models, the work provides a sound foundational introduction to the modeling and analysis of queueing systems for a broad interdisciplinary audience of students in mathematics, statistics, and applied disciplines such as computer science, operations research, and engineering. This edition includes additional topics in methodology and applications. Key features: • An introductory chapter including a historical account of the growth of queueing theory in more than 100 years. • A modeling-based approach with emphasis on identification of models • Rigorous treatment of the foundations of basic models commonly used in applications with appropriate references for advanced topics. • A chapter on matrix-analytic method as an alternative to the traditional methods of analysis of queueing systems. • A comprehensive treatment of statistical inference for queueing systems. • Modeling exercises and review exercises when appropriate. The second edition of An Introduction of Queueing Theory may be used as a textbook by first-year graduate students in fields such as computer science, operations research, industrial and systems engineering, as well as related fields such as manufacturing and communications engineering. Upper-level undergraduate students in mathematics, statistics, and engineering may also use the book in an introductory course on queueing theory. With its rigorous coverage of basic material and extensive bibliography of the queueing literature, the work may also be useful to applied scientists and practitioners as a self-study reference for applications and further research. "...This book has brought a freshness and novelty as it deals mainly with modeling and analysis in applications as well as with statistical inference for queueing problems. With his 40 years of valuable experience in teaching and high level research in this subject area, Professor Bhat has been able to achieve what he aimed: to make [the work] somewhat different in content and approach from other books." - Assam Statistical Review of the first edition

Probability, Statistics, and Stochastic Processes

Author : Peter Olofsson
Publisher : John Wiley & Sons
ISBN 13 : 0470889748
Total Pages : 573 pages
Book Rating : 4.4/5 (78 download)

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Book Synopsis Probability, Statistics, and Stochastic Processes by : Peter Olofsson

Download or read book Probability, Statistics, and Stochastic Processes written by Peter Olofsson and published by John Wiley & Sons. This book was released on 2012-05-22 with total page 573 pages. Available in PDF, EPUB and Kindle. Book excerpt: Praise for the First Edition ". . . an excellent textbook . . . well organized and neatly written." —Mathematical Reviews ". . . amazingly interesting . . ." —Technometrics Thoroughly updated to showcase the interrelationships between probability, statistics, and stochastic processes, Probability, Statistics, and Stochastic Processes, Second Edition prepares readers to collect, analyze, and characterize data in their chosen fields. Beginning with three chapters that develop probability theory and introduce the axioms of probability, random variables, and joint distributions, the book goes on to present limit theorems and simulation. The authors combine a rigorous, calculus-based development of theory with an intuitive approach that appeals to readers' sense of reason and logic. Including more than 400 examples that help illustrate concepts and theory, the Second Edition features new material on statistical inference and a wealth of newly added topics, including: Consistency of point estimators Large sample theory Bootstrap simulation Multiple hypothesis testing Fisher's exact test and Kolmogorov-Smirnov test Martingales, renewal processes, and Brownian motion One-way analysis of variance and the general linear model Extensively class-tested to ensure an accessible presentation, Probability, Statistics, and Stochastic Processes, Second Edition is an excellent book for courses on probability and statistics at the upper-undergraduate level. The book is also an ideal resource for scientists and engineers in the fields of statistics, mathematics, industrial management, and engineering.

Basic Probability Theory with Applications

Author : Mario Lefebvre
Publisher : Springer Science & Business Media
ISBN 13 : 0387749950
Total Pages : 340 pages
Book Rating : 4.3/5 (877 download)

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Book Synopsis Basic Probability Theory with Applications by : Mario Lefebvre

Download or read book Basic Probability Theory with Applications written by Mario Lefebvre and published by Springer Science & Business Media. This book was released on 2009-10-03 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main intended audience for this book is undergraduate students in pure and applied sciences, especially those in engineering. Chapters 2 to 4 cover the probability theory they generally need in their training. Although the treatment of the subject is surely su?cient for non-mathematicians, I intentionally avoided getting too much into detail. For instance, topics such as mixed type random variables and the Dirac delta function are only brie?y mentioned. Courses on probability theory are often considered di?cult. However, after having taught this subject for many years, I have come to the conclusion that one of the biggest problems that the students face when they try to learn probability theory, particularly nowadays, is their de?ciencies in basic di?erential and integral calculus. Integration by parts, for example, is often already forgotten by the students when they take a course on probability. For this reason, I have decided to write a chapter reviewing the basic elements of di?erential calculus. Even though this chapter might not be covered in class, the students can refer to it when needed. In this chapter, an e?ort was made to give the readers a good idea of the use in probability theory of the concepts they should already know. Chapter 2 presents the main results of what is known as elementary probability, including Bayes’ rule and elements of combinatorial analysis.