ch02 - $ Ch.2 1 Probability Introduction Probability theory...

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& $ % 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 finds a much wider applicability as it models a wide variety of chance phenomena . By chance phenomenon we mean any action, process or situation the outcome of which is random. An expression synonymous to chance phenomenon is probabilistic experiment . The chance phenomena we will deal with are observational studies and statistical experiments, including random sampling by any sampling method. For introducing probability concepts, and for demonstrating probability calculations, we will also
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& $ % 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 Sample Spaces The chance phenomena of relevance to statistics, for which probability theory provides models, include all
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& $ % statistical experiments and observational studies. As it was discussed in Chapter 1, to any statistical experiment or observational study there corresponds a population (or populations) of interest. The purpose of this section is to point out that the same is true also for the probabilistic experiments we will discuss. Moreover, we will see that it is often possible to associate both a hypothetical and a or non-hypothetical population to such experiments. 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 1 from the finite population of 52 cards. This finite population is the population that corresponds 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 1 from the finite population of the numbers { 1 , 2 , 3 , 4 , 5 , 6 } . Alternatively, rolling a die can be thought of as taking a simple random sample of size 1 from the infinite (and also hypothetical) population of all possible rolls of a die.
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& $ % Thus, either a finite or an infinite population can be thought of as the population corresponding to this probabilistic experiment. c) Consider the probabilistic experiment of rolling two dice. This can be thought of as obtaining a simple random sample of size 1 from the finite population of the pairs of numbers (1 , 1) , (1 , 2) , . . . , (1 , 6) (2 , 1) , (2 , 2) , . . . , (2 , 6) .
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