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A.2 SAMPLE SPACE, SAMPLE POINTS, AND EVENTS

The set of all possible outcomes of a random, or chance, experiment is called the population, or sample space, and each member of this sample space is called a sample point. Thus, in the experiment of tossing two coins, the sample space consists of these four possible outcomes: HH, HT, TH, and TT, where HH means a head on the first toss and also a head on the second toss, HT means a head on the first toss and a tail on the second toss, and so on. Each of the preceding occurrences constitutes a sample point.

An event is a subset of the sample space. Thus, if we let A denote the occurrence of one head and one tail, then, of the preceding possible outcomes, only two belong to A, namely HT and TH. In this case A constitutes an event. Similarly, the occurrence of two heads in a toss of two coins is an event. Events are said to be mutually exclusive if the occurrence of one event precludes the occurrence of another event. If in the preceding example HH occurs, the occurrence of the event HT at the same time is not possible. Events are said to be (collectively) exhaustive if they exhaust all the possible outcomes of an experiment. Thus, in the example, the events (a) two heads, (b) two tails, and (c) one tail, one head exhaust all the outcomes; hence they are (collectively) exhaustive events.

### A.3 PROBABILITY AND RANDOM VARIABLES Probability

Let A be an event in a sample space. By P(A), the probability of the event A, we mean the proportion of times the event A will occur in repeated trials of an experiment. Alternatively, in a total of possible equally likely out-

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comes of an experiment, if m of them are favorable to the occurrence of the event A, we define the ratio m/n as the relative frequency of A. For large values of n, this relative frequency will provide a very good approximation of the probability of A.

Properties of Probability. P(A) is a real-valued function1 and has these properties:

2. If A, B, C,... constitute an exhaustive set of events, then P(A + B + C + • • •) = 1, where A + B + C means A or B or C, and so forth.

3. If A, B, C,... are mutually exclusive events, then

P (A + B + C + ■■■) = P (A) + P (B) + P (C) + ... 