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

Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP

href="#fb3_img_img_5ddac734-b248-5ae9-b869-14c2d9a54966.png" alt="images"/> are all nonnegative, and their sum is 1. We may think of images

as observed weights or measures of occurrence of images

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 images

corresponding to the idealized values of images

, respectively, for an indefinitely large number of trials. It is assumed that images

are all positive numbers and that

(3.3.1) equation

The quantities images are called probabilities of occurrence of images , respectively.

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

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

It is evident, probabilities of events in a finite sample space S are values of an additive set function images 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 images is an infinite sequence of disjoint events, then

(3.3.2d) equation

As images and images are disjoint events, then from condition 3, we obtain

(3.3.3) equation

But since images and images , we have the following:

Theorem 3.3.1 (Rule of complementation) If E is an event in a sample space S, then

(3.3.4) equation

The law of complementation provides a simple method of finding the probability of an event images , if E is an event whose probability is easy to find. We sometimes say that the odds in favor of E are

(3.3.4a) equation

which from 3.3.4 takes the form images . The reader may note that images .

Example 3.3.1 (Tossing coins) Suppose that 10 coins are tossed and we ask for the probability of getting at least 1 head. In this example, the sample space S has images sample points. If the coins are unbiased, the sample points are equally likely (sample points are called equally likely if each sample point has the same probability of occurring), so that to each of the sample points the probability images is assigned. If we denote by E the event of getting no heads, then E contains only one sample point, and images , of course, has 1023 sample points. Thus

The odds on E and images are clearly images and images .

Referring to the statement in Theorem 3.3.1 that images and images are disjoint events whose union is S, we have the following rule.

Theorem 3.3.2 (General rule of complementation) If images are events in a sample space S, then we have

(3.3.5) equation

Another useful result follows readily from (3.3.2c) by mathematical induction

Theorem 3.3.3 (Rule of addition of probabilities for mutually exclusive events) If images are disjoint events in a sample space S, then

(3.3.6) equation

Example 3.3.2 (Determination of probabilities

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