CHAPTER 2 PROBABILITY

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depends on the objectives of the study. As speci c questions occur later in the book, appropriate sample spaces are discussed. EXAMPLE 2-2 If two connectors are selected and measured, the extension of the positive real line R is to take the sample space to be the positive quadrant of the plane: S R R

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If the objective of the analysis is to consider only whether or not the parts conform to the manufacturing speci cations, either part may or may not conform. We abbreviate yes and no as y and n. If the ordered pair yn indicates that the rst connector conforms and the second does not, the sample space can be represented by the four outcomes: S 5yy, yn, ny, nn6

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If we are only interested in the number of conforming parts in the sample, we might summarize the sample space as S 50, 1, 26

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As another example, consider an experiment in which the thickness is measured until a connector fails to meet the speci cations. The sample space can be represented as S 5n, yn, yyn, yyyn, yyyyn, and so forth6

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In random experiments in which items are selected from a batch, we will indicate whether or not a selected item is replaced before the next one is selected. For example, if the batch consists of three items {a, b, c} and our experiment is to select two items without replacement, the sample space can be represented as Swithout 5ab, ac, ba, bc, ca, cb6

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This description of the sample space maintains the order of the items selected so that the outcome ab and ba are separate elements in the sample space. A sample space with less detail only describes the two items selected {{a, b}, {a, c}, {b, c}}. This sample space is the possible subsets of two items. Sometimes the ordered outcomes are needed, but in other cases the simpler, unordered sample space is suf cient. If items are replaced before the next one is selected, the sampling is referred to as with replacement. Then the possible ordered outcomes are Swith 5aa, ab, ac, ba, bb, bc, ca, cb, cc6

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The unordered description of the sample space is {{a, a}, {a, b}, {a, c}, {b, b}, {b, c}, {c, c}}. Sampling without replacement is more common for industrial applications. Sometimes it is not necessary to specify the exact item selected, but only a property of the item. For example, suppose that there are 5 defective parts and 95 good parts in a batch. To study the quality of the batch, two are selected without replacement. Let g denote a good part and d denote a defective part. It might be suf cient to describe the sample space (ordered) in terms of quality of each part selected as S 5gg, gd, dg, dd6

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2-1 SAMPLE SPACES AND EVENTS

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Message 1 on time Message 2 on time Message 3 late on time late late

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Figure 2-5 Tree diagram for three messages.

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One must be cautious with this description of the sample space because there are many more pairs of items in which both are good than pairs in which both are defective. These differences must be accounted for when probabilities are computed later in this chapter. Still, this summary of the sample space will be convenient when conditional probabilities are used later in this chapter. Also, if there were only one defective part in the batch, there would be fewer possible outcomes S 5gg, gd, dg6

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because dd would be impossible. For sampling questions, sometimes the most important part of the solution is an appropriate description of the sample space. Sample spaces can also be described graphically with tree diagrams. When a sample space can be constructed in several steps or stages, we can represent each of the n1 ways of completing the rst step as a branch of a tree. Each of the ways of completing the second step can be represented as n2 branches starting from the ends of the original branches, and so forth. EXAMPLE 2-3 Each message in a digital communication system is classi ed as to whether it is received within the time speci ed by the system design. If three messages are classi ed, use a tree diagram to represent the sample space of possible outcomes. Each message can either be received on time or late. The possible results for three messages can be displayed by eight branches in the tree diagram shown in Fig. 2-5. An automobile manufacturer provides vehicles equipped with selected options. Each vehicle is ordered With or without an automatic transmission With or without air-conditioning With one of three choices of a stereo system With one of four exterior colors

EXAMPLE 2-4

If the sample space consists of the set of all possible vehicle types, what is the number of outcomes in the sample space The sample space contains 48 outcomes. The tree diagram for the different types of vehicles is displayed in Fig. 2-6. EXAMPLE 2-5 Consider an extension of the automobile manufacturer illustration in the previous example in which another vehicle option is the interior color. There are four choices of interior color: red, black, blue, or brown. However, With a red exterior, only a black or red interior can be chosen. With a white exterior, any interior color can be chosen.