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Unformatted text preview: Ch.2 Probability 1 Introduction Probability theory arose from the need to quantify the likelihood of occurrence of certain events associated with games of chance. Today, probability theory is applied much more widely as it models a wide variety of chance phenomena . By chance phenomena we mean any actions, processes or situations the outcomes of which are random. Thus, kicking a soccer ball, and stock market fluctuations, are both chance phenomena. An expression synonymous to chance phenomena is probabilistic or random experiments . The random experiments we will deal with are observational studies and statistical ex periments, including the random sampling methods discussed in Chapter 1. To introduce probability concepts, and to demonstrate probability calculations, we will also talk about such probabilistic experiments as picking a card from a deck, or rolling a die. In this chapter we introduce, at an elementary level, the basic concepts of probability theory, including conditional probability and the notion of independence, and describe common techniques for calculating probabilities. This, and the specific probability models which will be discussed in Chapters 3, 4 and 5 will provide the needed probabilistic background for discussing statistical inference. In this chapter the term ’experiment’ will be used in its wider sense, i.e. to indicate a probability experiment. 2 Probabilistic Experiments, Populations, and Sam ple Spaces As we saw in Chapter 1, to any sampling experiment there corresponds a population. A distinction was made between the population of units and the statistical population. The purpose of this section is to point out that the random experiments we will consider can be thought of, equivalently, as sampling experiments from some population. In fact, we will see for each random experiment we can conceptualize several equivalent sampling experiments. One of the different ways involves the concept of sample space which we will introduce. The following simple examples illustrating these points. 1 EXAMPLE A: a) Consider the probabilistic experiment of picking a card from a deck of cards. This can be thought of as obtaining a simple random sample of size one from the infinite hypothetical population of all selections of one card from a deck of cards. Alternatively, it can be thought of as obtaining a simple random sample of size 1 from the finite population of 52 cards. Thus, either a finite or an infinite population can be thought of as the population corresponding to this probabilistic experiment. b) Consider the probabilistic experiment of rolling a die. This can be thought of as obtaining a simple random sample of size one from the infinite hypothetical population of all possible rolls of a die. Alternatively, rolling a die can be thought of as taking a simple random sample of size one from the finite population of the numbers { 1 , 2 , 3 , 4 , 5 , 6 } . Thus, either a finite or an infinite population can be thought of as the population corresponding...
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 Fall '00
 Akritas
 Probability, Probability theory

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