Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP. Bhisham C. Gupta

Figure 3.2.3 Venn diagram representing events

Similarly the event consisting of all elements contained in all is the intersection of written as

      (3.2.9)

      If for every pair of events (), , from we have that , then are disjoint and mutually exclusive events.

      An important result concerning several events is the following theorem.

      Theorem 3.2.1 If are events in a sample space S, then and are disjoint events whose union is S.

      This result follows by noting that the events and are complement of each other.

3.3 Concepts of Probability

      Suppose that a sample space S, consists of a finite number, say m, of elements , so that the elements are such that for all and also represent an exhaustive list of outcomes in S, so that . If the operation whose sample space is S is repeated a large number of times, some of these repetitions will result in , some in , and so on. (The separate repetitions are often called trials.) Let be the fractions of the total number of trials resulting in , respectively. Then, are all nonnegative, and their sum is 1. We may think of as observed weights or measures of occurrence of obtained on the basis of an experiment consisting of a large number of repeated trials. If the entire experiment is repeated, another set of f's would occur with slightly different values, and so on for further repetitions. If we think of indefinitely many repetitions, we can conceive of idealized values being obtained for the f's. It is impossible, of course, to show that in a physical experiment, the f's converge to limiting values, in a strict mathematical sense, as the number of trials increases indefinitely. So we postulate values corresponding to the idealized values of , respectively, for an indefinitely large number of trials. It is assumed that are all positive numbers and that

      (3.3.1)

      The quantities are called probabilities of occurrence of , respectively.

Now suppose that E is any event in S that consists of a set of one or more e's, say . Thus . The probability of the occurrence of E is denoted by and is defined as follows:

      If E contains only one element, say , it is written as

      It is evident, probabilities of events in a finite sample space S are values of an additive set function defined on sets E in S, satisfying the following conditions:

      1 If E is any event in S, then(3.3.2a)

      2 If E is the sample space S itself, then(3.3.2b)

      3 If E and F are two disjoint events in S, then(3.3.2c)

      These conditions are also sometimes known as axioms of probability. In the case of an infinite sample space S, condition 3 extends as follows:

      if is an infinite sequence of disjoint events, then

      (3.3.2d)

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