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Ash_Book_Ch1_Basic+Probability

# Ash_Book_Ch1_Basic+Probability - CHAPTER 1 Basic...

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CHAPTER Basic Probability 1 SECTION 1 -1 PROBABILITY SPACES We want to answer the questions "What are probabilities?" and "How does an event get a probability?" Sample Space of an Experiment A sample space corresponding to an experiment is a set of outcomes such that exactly one of the outcomes occurs when the experiment is performed. The sample space is often called the universe, and the outcomes are called points in the sample space. There is more than one way to view an experiment, so an experiment can have more than one associated sample space. For example, suppose you draw one card from a deck. Here are some sample spaces. sample space 1 (the most popular) The space consists of 52 outcomes, 1 for each card in the deck. sample space 2 This space consists of just 2 outcomes, black and red. sample space 3 This space consists of 13 outcomes, namely, 2,3,4,..., 10, J, Q, K, A. sample space 4 This space consists of 2 outcomes, picture and non- picture. Any outcome or collection of outcomes in a sample space is called an event, including the null (empty) set of outcomes and the set of all outcomes. In the first sample space, "black" is an event (consisting of 26 points). It is also an event in sample space 2 (consisting of 1 point). It is not an event in 1

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2 Sec. 1-1 Probability Spaces sample spaces 3 and 4, so these spaces are not useful if you are interested in the outcome black. Similarly, "king" is an event in sample spaces 1 and 3 but not in 2 and 4. Probability Spaces Consider a sample space with n points. Probabilities are numbers assigned to events satisfying the following rules. (1) Each outcome is assigned a non-negative probability such that the sum of the n probabilities is 1. This axiom corresponds to our intuitive understanding of probabilities in real life. The weather reporter never predicts a negative chance of snow, and the chance of snow plus the chance of rain plus the chance of dry should be 100%, that is, 1, (2) If A is an event and P(A) denotes the probability of A, then P(A) = sum of the probabilities of the outcomes in the event A A sample space together with an assignment of probabilities to events is called a probability space. Note that probabilities are always between 0 and 1. Figure 1 shows a probability space with six outcomes a, b, c, d, e, / and their respective probabilities. The indicated event B contains the three out- comes d, e, / and P(B) = .l + .2 + .3 = .6 Figure 1 Probabilities may be initially assigned to outcomes any way you like, as long as (1) is satisfied. Then probabilities of events are determined by (2). To make our probabilities useful, we try to assign initial probabilities to make a "good" model for the experiment. h .2 a A B d .1 /' .3 c .1 e .2
Chap. 1 Basic Probability 3 Suppose you put slips of paper labeled a, h, c, d, e, / in a bag, shake it thoroughly, put on a blindfold, and draw one out; that is, you pick one of a, 6, c, d, e, / at random. The appropriate model should have the six outcomes equally likely, so instead of the probabilities in Fig. 1, you should assign each outcome the probability 1/6. Then 1 1 1 3

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Ash_Book_Ch1_Basic+Probability - CHAPTER 1 Basic...

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