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Chapt 1a

# Chapt 1a - 1 Basic Probability Concepts 1.1 1.2 1.3 1.4 1.5...

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1 Basic Probability Concepts 1.1 Introduction 1 1.2 Sample Space and Events 2 1.3 Definitions of Probability 4 1.4 Applications of Probability 7 1.5 Elementary Set Theory 8 1.6 Properties of Probability 13 1.7 Conditional Probability 15 1.8 Independent Events 26 1.9 Combined Experiments 29 1.10 Basic Combinatorial Analysis 31 1.11 Reliability Applications 42 1.12 Chapter Summary 47 1.13 Problems 47 1.14 References 57 1.1 Introduction Probability deals with unpredictability and randomness, and probability theory is the branch of mathematics that is concerned with the study of random phe- nomena. A random phenomenon is one that, under repeated observation, yields different outcomes that are not deterministically predictable. However, these outcomes obey certain conditions of statistical regularity whereby the relative frequency of occurrence of the possible outcomes is approximately predictable. Examples of these random phenomena include the number of electronic mail (e-mail) messages received by all employees of a company in one day, the num- ber of phone calls arriving at the university’s switchboard over a given period, 1

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2 Chapter 1 Basic Probability Concepts the number of components of a system that fail within a given interval, and the number of A’s that a student can receive in one academic year. According to the preceding definition, the fundamental issue in random phe- nomena is the idea of a repeated experiment with a set of possible outcomes or events. Associated with each of these events is a real number called the proba- bility of the event that is related to the frequency of occurrence of the event in a long sequence of repeated trials of the experiment. In this way it becomes obvi- ous that the probability of an event is a value that lies between zero and one, and the sum of the probabilities of the events for a particular experiment should sum to one. This chapter begins with events associated with a random experiment. Then it provides different definitions of probability and considers elementary set theory and algebra of sets. Finally, it discusses basic concepts in combinatorial analysis that will be used in many of the later chapters. 1.2 Sample Space and Events The concepts of experiments and events are very important in the study of prob- ability. In probability, an experiment is any process of trial and observation. An experiment whose outcome is uncertain before it is performed is called a random experiment. When we perform a random experiment, the collection of possible elementary outcomes is called the sample space of the experiment, which is usu- ally denoted by S . We define these outcomes as elementary outcomes because exactly one of the outcomes occurs when the experiment is performed. The ele- mentary outcomes of an experiment are called the sample points of the sample space and are denoted by w i , i = 1 , 2 ,... . If there are n possible outcomes of an experiment, then the sample space is S = { w 1 , w 2 ,..., w n } .
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Chapt 1a - 1 Basic Probability Concepts 1.1 1.2 1.3 1.4 1.5...

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