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An event is a subset A of the sample space S, i.e., it is a set of possible outcomes. If the outcome of an experiment is an element of A, we say that the event A has occurred. An event consisting of a single point of S is often called a simple or elementary event.
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EXAMPLE 1.8 If we toss a coin twice, the event that only one head comes up is the subset of the sample space that consists of points (0, 1) and (1, 0), as indicated in Fig. 1-2.
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Fig. 1-2
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As particular events, we have S itself, which is the sure or certain event since an element of S must occur, and the empty set \ , which is called the impossible event because an element of \ cannot occur. By using set operations on events in S, we can obtain other events in S. For example, if A and B are events, then 1. 2. 3. 4. A < B is the event either A or B or both. A < B is called the union of A and B. A d B is the event both A and B. A d B is called the intersection of A and B. Ar is the event not A. Ar is called the complement of A. A B A d Br is the event A but not B. In particular, Ar S A.
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If the sets corresponding to events A and B are disjoint, i.e., A d B \ , we often say that the events are mutually exclusive. This means that they cannot both occur. We say that a collection of events A1, A2, c , An is mutually exclusive if every pair in the collection is mutually exclusive.
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EXAMPLE 1.9 Referring to the experiment of tossing a coin twice, let A be the event at least one head occurs and B the event the second toss results in a tail. Then A {HT, TH, HH }, B {HT, TT }, and so we have
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5HT, TH, HH, TT 6 Ar 5TT 6 A B
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A>B 5TH, HH 6
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CHAPTER 1 Basic Probability
The Concept of Probability
In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is convenient to assign a number between 0 and 1. If we are sure or certain that the event will occur, we say that its probability is 100% or 1, but if we are sure that the event will not occur, we say that its probability is zero. If, for example, the prob1 ability is 4, we would say that there is a 25% chance it will occur and a 75% chance that it will not occur. Equivalently, we can say that the odds against its occurrence are 75% to 25%, or 3 to 1. There are two important procedures by means of which we can estimate the probability of an event. 1. CLASSICAL APPROACH. If an event can occur in h different ways out of a total number of n possible ways, all of which are equally likely, then the probability of the event is h > n.
EXAMPLE 1.10 Suppose we want to know the probability that a head will turn up in a single toss of a coin. Since there are two equally likely ways in which the coin can come up namely, heads and tails (assuming it does not roll away or stand on its edge) and of these two ways a head can arise in only one way, we reason that the required probability is 1 > 2. In arriving at this, we assume that the coin is fair, i.e., not loaded in any way.
2. FREQUENCY APPROACH. If after n repetitions of an experiment, where n is very large, an event is observed to occur in h of these, then the probability of the event is h > n. This is also called the empirical probability of the event.
EXAMPLE 1.11 If we toss a coin 1000 times and find that it comes up heads 532 times, we estimate the probability of a head coming up to be 532 > 1000 0.532.
Both the classical and frequency approaches have serious drawbacks, the first because the words equally likely are vague and the second because the large number involved is vague. Because of these difficulties, mathematicians have been led to an axiomatic approach to probability.
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