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E1 E2 E4 E3
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Figure 2-11 Venn diagram of four mutually exclusive events.
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Upon expanding P1A B2 by Equation 2-1 and using the distributed rule for set operations to simplify P3 1A B2 C4 , we obtain P1A B C2 P1A2 P1A2 P1A2 P1B2 P1B2 P1B2 P1A B2 P1C2 P1A B2 P1C2 3P1A C2 P1B P1C2 P1A B2 P1B C2 C2 P1A B C2 4 P1A C2 P1A B C2 P3 1A C2 1B C2 4
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We have developed a formula for the probability of the union of three events. Formulas can be developed for the probability of the union of any number of events, although the formulas become very complex. As a summary, for the case of three events
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P1A B C2
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Results for three or more events simplify considerably if the events are mutually exclusive. In general, a collection of events, E1, E2, p , Ek , is said to be mutually exclusive if there is no overlap among any of them. The Venn diagram for several mutually exclusive events is shown in Fig. 2-11. By generalizing the reasoning for the union of two events, the following result can be obtained:
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A collection of events, E1, E2, p , Ek, is said to be mutually exclusive if for all pairs, Ei Ej For a collection of mutually exclusive events, P1E1 E2 p Ek 2 P1E1 2 P1E2 2 p P1Ek 2 (2-4)
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EXAMPLE 2-15
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A simple example of mutually exclusive events will be used quite frequently. Let X denote the pH of a sample. Consider the event that X is greater than 6.5 but less than or equal to 7.8. This
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CHAPTER 2 PROBABILITY
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probability is the sum of any collection of mutually exclusive events with union equal to the same range for X. One example is P16.5 X 7.82 P16.5 X 7.02 P17.0 X 7.52 P17.5 X 7.82
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Another example is P16.5 X 7.82 P16.5 X 6.62 P16.6 X 7.12 P17.1 X 7.42 P17.4
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The best choice depends on the particular probabilities available.
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EXERCISES FOR SECTION 2-3
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2-49. If P 1A2 0.3 , P 1B2 0.2, and P 1A B2 0.1, determine the following probabilities: (a) P1A 2 (b) P1A B2 (c) P1A B2 (d) P1A B 2 (e) P3 1A B2 4 (f) P1A B2 2-50. If A, B, and C are mutually exclusive events with P 1A2 0.2, P 1B2 0.3, and P 1C 2 0.4, determine the following probabilities: (a) P1A B C2 (b) P1A B C2 (c) P1A B2 (d) P3 1A B2 C 4 (e) P1A B C 2 2-51. If A, B, and C are mutually exclusive events, is it possible for P(A) 0.3, P(B) 0.4, and P(C) 0.5 Why or why not 2-52. Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: shock resistance high low 70 9 16 5 (a) If a shaft is selected at random, what is the probability that the shaft conforms to surface nish requirements (b) What is the probability that the selected shaft conforms to surface nish requirements or to roundness requirements (c) What is the probability that the selected shaft either conforms to surface nish requirements or does not conform to roundness requirements (d) What is the probability that the selected shaft conforms to both surface nish and roundness requirements 2-54. Cooking oil is produced in two main varieties: monoand polyunsaturated. Two common sources of cooking oil are corn and canola. The following table shows the number of bottles of these oils at a supermarket: type of oil canola corn 7 13 93 77
type of unsaturation
mono poly
scratch resistance
high low
(a) If a disk is selected at random, what is the probability that its scratch resistance is high and its shock resistance is high (b) If a disk is selected at random, what is the probability that its scratch resistance is high or its shock resistance is high (c) Consider the event that a disk has high scratch resistance and the event that a disk has high shock resistance. Are these two events mutually exclusive 2-53. The analysis of shafts for a compressor is summarized by conformance to speci cations. roundness conforms yes no 345 5 12 8
(a) If a bottle of oil is selected at random, what is the probability that it belongs to the polyunsaturated category (b) What is the probability that the chosen bottle is monounsaturated canola oil 2-55. A manufacturer of front lights for automobiles tests lamps under a high humidity, high temperature environment using intensity and useful life as the responses of interest. The following table shows the performance of 130 lamps: useful life satisfactory unsatisfactory intensity satisfactory 117 3 unsatisfactory 8 2 (a) Find the probability that a randomly selected lamp will yield unsatisfactory results under any criteria. (b) The customers for these lamps demand 95% satisfactory results. Can the lamp manufacturer meet this demand 2-56. The shafts in Exercise 2-53 are further classi ed in terms of the machine tool that was used for manufacturing the shaft.