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Solution Manual for Applied Statistics and Probability for Engineers, 7th Edition

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 1 CHAPTER 2 Section 2 - 1 Provide a reasonable description of the sample space for each of the random experiments in Exercises 2 .1.1 to 2.1.11 . There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make. 2.1.1 . Each of four transmitted bits is classified as either in error or not in error. Let e and o denote a bit in error and not in error ( o denotes okay) , respectively.                oooo oeoo eooo eeoo oooe oeoe eooe eeoe ooeo oeeo eoeo eeeo ooee oeee eoee eeee S , , , , , , , , , , , , , , , 2.1.2 . The number of hits (views) is recorded at a high - volume Web site in a day.   ,... 2 , 1 , 0  S = set of nonnegative integers 2.1.3 . In the final inspection of electronic power supplies, either units pass, or three types of nonconformities might occur: functional, minor, or cosmetic. Three units are inspected. Let a denote an acceptable power supply. Let f , m , and c denote a power supply that has a functional, minor, or cosmetic error, respectively.   c m f a S , , ,  2.1.4 . An ammeter that displays three digits is used to measure current in milliamperes. A vector with three components can describe the three digits of the ammeter. Each digit can be 0,1,2,. ..,9. The sample space S is 1000 possible three digit integers,   999 ,..., 001 , 000  S 2.1.5 . The following two questions appear on an employee survey questionnaire. Each answer is chosen from the five point scale 1 (never), 2, 3, 4, 5 (always). Is the corporation willing to listen to and fairly evaluate new ideas? How often are my coworkers important in my overall job performance? Let an ordered pair of numbers, such as 43 denote the response on the first and second question. Then, S consists of the 25 ordered pairs   1112 55 , ,..., 2.1.6 . The time until a service transaction is requested of a computer to the nearest millisecond.   ,..., 2 , 1 , 0  S in milliseconds 2.1.7 . The pH reading of a water sample to the nearest tenth of a unit.   0 . 14 , 2 . 1 , 1 . 1 , 0 . 1   S 2.1.8 . The voids in a ferrite slab are classified as small, medium, or large. The number of voids in each category is measured by an optical inspection of a sample. Let s , m , and l denote small, medium, and large, respectively. Then S = { s , m , l , ss , sm , sl , ….} 2.1.9 . A sampled injection - molded part could have been produced in either one of two presses and in any one of the eight

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 2 cavities in each press. PRESS CAVITY 1 2 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 2.1.10 . An order for an automobile can specify either an automatic or a standard transmission, either with or without air conditioning, and with any one of the four colors red, blue, black, or white. Describe the set of possible orders for this experiment. automatic transmission transmission standard without with air without air air air with white red blue black white red blue black white red blue black white red blue black 2.1.11 . Calls are repeatedly placed to a busy phone line until a connection is achieved. Let c and b denote connect and busy, respectively. Then S = { c , bc , bbc , bbbc , bbbbc , …} 2.1.12 . Three attempts are made to read data in a magnetic storage device before an error recovery procedure that repositions the magnetic head is used. The error recovery procedure attempts three repositionings before an “abort’’ message is sent to the operator. Let s denote the success of a read operation f denote the failure of a read operation S denote the success of an error recovery procedure F denote the failure of an error recovery procedure A denote an abort message sent to the operator Describe the sample space of this experiment with a tree diagram.   fffFFFA fffFFS fffFS fffS ffs fs s S , , , , , , 

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 3 2.1.13 . Three events are shown on the Venn diagram in the following figure: Reproduce the fi gure and shade the region that corresponds to each of the following events. (a) A  (b) A  B (c)  A  B   C (d)  B  C   (e)  A  B    C ( a) (b) ( c) ( d)

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 4 ( e) 2.1.14 In an injection - molding operation, the length and width, denoted as X and Y , respectively, of each molded part are evaluated. Let A denote the event of 48 < X < 52 centimeters B denote the event of 9 < Y < 11 centimeters Construct a Venn diagram that includes these events. Shade the areas that represent the following: (a) A (b) A  B (c) A   B (d) A  B (e) If these events were mutually exclusive, how successful would this production operation be? Would the process produce parts with X  50 centimeters and Y = 10 centimeters? ( a) ( b) A B 48 52 11 9 A B 48 52 11 9

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 5 ( c) ( d) (e) If the events are mutually exclusive, then A  B is the null set. Therefore, the process does not produce product parts with X = 50 cm and Y = 1 0 cm. The process would not be successful. 2.1.15 . A digital scale that provides weights to the nearest gram is used. (a) What is the sample space for this experiment? Let A denote the event that a weight exceeds 11 grams, let B denote the event that a weight is less than or equal to 15 grams, and let C denote the event that a weight is greater than or equal to 8 grams and less than 12 grams. Describe the following events. (b) A  B (c) A  B (d) A  (e) A  B  C (f)  A  C   (g) A  B  C (h) B   C (i) A   B  C  ( a) Let S = the nonnegative integers from 0 to the largest integer that can be displayed by the scale. Let X denote the weight. A is the event that X > 11 B is the event that X  15 C is the event that 8  X <12 S = {0, 1, 2, 3, …} ( b) S ( c) 11 < X  15 or {12, 13, 14, 15} ( d) X  11 or {0, 1, 2, …, 11} ( e) S ( f) A  C contain s the values of X such that: X  8 Thus ( A  C )  contain s the values of X such that: X < 8 or {0, 1, 2, …, 7} ( g)  ( h) B  contain s the values of X such that X > 15. Therefore, B   C is the empty set. They have no outcomes in common or  . ( i) B  C is the event 8  X <12. Therefore, A  ( B  C ) is the event X  8 or {8, 9, 10, …} 2.1.16. In light - dependent photosynthesis, light quality refers to the wavelengths of light that are important. The wavelength of a sample of photosynthetically active radiations (PAR) is measured to the nearest nanometer. The red range is 675 – 700 nm and the blue range is 450 – 500 nm. Let A denote the event that PAR occurs in the red range, and let B denote the event that PAR occurs in the blue range. Describe the sample space and indicate each of the following events: A B 48 52 11 9 A B 48 52 11 9

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 6 (a) A (b) B (c) A  B (d) A  B Let w denote the wavelength. The sample space is { w | w = 0, 1, 2, …} (a) A ={ w | w = 675, 676, …, 700 nm } (b) B ={ w | w = 450, 451, …, 500 nm } (c)    B A (d)   B A { w | w = 450, 451, …, 500, 675, 676, …, 700 nm} 2.1.17 . Four bits are transmitted over a digital communications channel. Each bit is either distorted or received without distortion. Let Ai denote the event that the i th bit is distorted, i  1 ,        , 4. (a) Describe the sample space for this experiment. (b) Are the A i ’s mutually exclusive? Describe the outcomes in each of the following events: (c) A 1 (d) A 1  (e) A 1  A 2  A 3  A 4 (f)  A 1  A 2    A 3  A 4  Let d and o denote a distorted bit and one that is not distorted ( o denotes okay), respectively. ( a)                oooo odoo dooo ddoo oood odod dood ddod oodo oddo dodo dddo oodd oddd dodd dddd S , , , , , , , , , , , , , , , ( b) No, for example   ddoo ddod dddo dddd A A , , , 2 1   ( c)                dooo ddoo dood ddod dodo dddo dodd dddd A , , , , , 1 ( d)                 oooo odoo oood odod oodo oddo oodd oddd A , , , , , , , 1 ( e) } { 4 3 2 1 dddd A A A A     ( f)   ddoo oodd ddod oddd dddo dodd dddd A A A A , , , , , , ) ( ) ( 4 3 2 1     2.1.18. Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized here:

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 7 Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. Determine the number of disks in A  B, A  , and A  B . A  B = 70, A  = 14, A  B = 95 2.1.19 . In control replication, cells are replicated over a period of two days. Not until mitosis is completed can freshly synthesized DNA be replicated again. Two control mechanisms have been identified — one positive and one negative. Suppose that a replication is observed in three cells. Let A denote the event that all cells are identified as positive, and let B denote the event that all cells are negative. Describe the sample space graphically and display each of the following events: (a) A (b) B (c) A  B (d) A  B Let P and N denote positive and negative, respectively. The sample space is { PPP , PPN , PNP , NPP , PNN , NPN , NNP , NNN }. (a) A ={ PPP } (b) B ={ NNN } (c)    B A (d)   B A { PPP , NNN } 2.1.20. Samples of emissions from three suppliers are classified for conformance to air - quality specifications. The results from 100 samples are summarized as follows: Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications. Determine the number of samples in A   B, B  and in A  B . B A   = 55, B  =23, B A  = 85 2.1.21 . The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space be positive, real numbers. Define the events A and B as follows: A   x | x  72  5  and B   x | x  52  5  . Describe each of the following events: (a) A  (b) B  (c) A  B (d) A  B ( a) A  = { x | x  72.5} ( b) B  = { x | x  52.5} ( c) A  B = { x | 52.5 < x < 72.5}

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 8 ( d) A  B = { x | x > 0} 2 .1.22 . The following table summarizes 204 endothermic reactions involving sodium bicarbonate. Let A denote the event that a reaction’s final temperature is 271 K or less. Let B denote the event that the heat absorbed is below target. Determine the number of reactions in each of the following events. (a) A  B (b) A  (c) A  B (d) A  B  (e) A   B  (a) A  B = 56 (b) A  = 36 + 56 = 92 (c) A  B = 40 + 12 + 16 + 44 + 56 = 168 ( d) A  B  = 40+12+16+44+36=148 (e) A   B  = 36 2.1.23. A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. How many different designs are possible? Total number of possible designs = 900 5 3 5 3 4      2.1.24. Consider the hospital emergency department data given below. Let A denote the event that a visit is to hospital 1, and let B denote the event that a visit results in admittance to any hospital. Determine the number of persons in each of the following events. (a) A  B (b) A  (c) A  B (d) A  B  (e) A   B  (a) A  B = 1277 (b) A  = 22252 – 5292 = 16960 (c) A  B = 1685 + 3733 + 1403 + 2 + 14 + 29 + 46 + 3 = 6915 (d) A  B  = 195 + 270 + 246 + 242+ 3820 + 5163 + 4728 + 3103 + 1277 = 19044 (e) A   B  = 270 + 246 + 242 + 5163 + 4728 + 3103 = 13752 2.1.25 . The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of Chronic Hepatitis C,” [ Gastroenterology (1996, Vol. 111, no. 5, pp. 1307 – 1312)], considered the effect of two treatments and a control for treatment of hepatitis C. The following table provides the total patients in each group and the number that showed a complete (positive) response after 24 we eks of treatment.

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 9 Let A denote the event that the patient was treated with ribavirin plus interferon alfa, and let B denote the event that the response was complete. Determine the number of patients in each of the following events. (a) A (b) A  B (c) A  B (d) A   B  Let |A| denote the number of elements in the set A. (a) |A| = 21 (b) |A∩B| = 16 (c) |A ⋃ B| = A+B - (A∩B) = 21+22 – 16 = 27 (d) |A'∩B'| = 60 - |AUB| = 60 – 27 = 33 2.1.26 . A computer system uses passwords that contain exactly eight characters, and each character is 1 of the 26 lowercase letters ( a – z ) or 26 uppercase letters ( A – Z ) or 10 integers (0 – 9). Let  denote the set of all possible passwords, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Determine the number of passwords in each of the following events. (a)  (b) A (c) A   B  (d) Passwords that contain at least 1 integer (e) Passwords that contain exactly 1 integer Let |A| denote the number of elements in the set A. (a) The number of passwords in  is    = 62 8 (from multiplication rule). (b) The number of passwords in A is |A|= 52 8 (from multiplication rule) (c) A' ∩ B' = (A U B)'. Also, |A| = 52 8 and |B| = 10 8 and A ∩ B = null. Therefore, (A U B)' =             62 8 - 52 8 - 10 8 ≈1.65 x 10 14 (d) Passwords that contain at least 1 integer = |  | - |A| = 62 8 – 52 8 ≈ 1.65 x 10 14 (e) Passwords that contain exactly 1 integer. The number of passwords with 7 letters is 52 7 . Also, 1 integer is selected in 10 ways, and can be inserted into 8 positions in the password. Therefore, the solution is 8(10)(52 7 ) ≈ 8.22 x 10 13 Section 2 - 2 2 .2.1. A sample of two printed circuit boards is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches: (a) The batch contains 90 boards that are not defective, 8 boards with minor defects, and 2 boards with major defects. (b) The batch contains 90 boards that are not defective, 8 boards with minor defects, and 1 board with major defects. Let g denote a good board, m a board with minor defects, and j a board with major defects. ( a) S = { gg, gm, gj, mg, mm, mj, jg, jm, jj } ( b) S ={ gg,gm,gj,mg,mm,mj,jg,jm } 2.2.2 . A sample of two items is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches: (a) The batch contains the items { a , b , c , d }. (b) The batch contains the items { a , b , c , d , e , f , g }. (c) The batch contains 4 defective items and 20 good items. (d) The batch contains 1 defective item and 20 good items. ( a) { ab, ac, ad, bc, bd, cd, ba, ca, da, cb, db, dc }

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 10 ( b) { ab, ac, ad, ae, af, ag, ba, bc, bd, be, bf, bg, ca, cb, cd, ce, cf, cg, da, db, dc, de, df, dg, ea, eb, ec, ed, ef, eg, fa, fb, fc, fg, fd, fe, ga, gb, gc, gd, ge, gf } , contains 42 elements ( c) Let d and g denote defective and good , respectively. Then S = { gg, gd, dg, dd } ( d) S = { gd, dg, gg } 2.2.3 . A wireless garage door opener has a code determined by the up or down setting of 12 switches. How many outcomes are in the sample space of possible codes? 2 12 = 4096 2.2.4 . In a manufacturing operation, a part is produced by machining, polishing, and painting. If there are three machine tools, four polishing tools, and three painting tools, how many different routings (consisting of machining, followed by polishing, and followed by painting) for a part are possible? From the multiplication rule, 3 4 3 36    2 .2.5 . New designs for a wastewater treatment tank have proposed three possible shapes, four possible sizes, three locations for input valves, and four locations for output valves. How many different product designs are possible? From the multiplication rule, 3 4 3 4 144     2.2.6. A manufacturing process consists of 10 operations that can be completed in any order. How many different production sequences are possible? From equation 2 .1 , the answer is 10! = 3 , 628 , 800 2.2.7 . A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. Assume 10 of the chips do not conform to customer requirements. (a) How many different samples are possible? (b) How many samples of five contain exactly one nonconforming chip? (c) How many samples of five contain at least one nonconforming chip? ( a) From equation 2 - 4, the number of samples of size five is   528 , 965 , 416 ! 135 ! 5 ! 140 140 5   ( b) There are 10 ways of selecting one nonconforming chip and there are   880 , 358 , 11 ! 126 ! 4 ! 130 130 4   ways of selecting four conforming chips. Therefore, the number of samples that contain exactly one nonconforming chip is 10   800 , 588 , 113 130 4   ( c) The number of samples that contain at least one nonconforming chip is the total number of samples   5 140 minus the number of samples that contain no nonconforming chips   5 130 . That is   5 140 -   5 130 = 752 , 721 , 130 ! 125 ! 5 ! 130 ! 135 ! 5 ! 140   2 .2.8 . In a sheet metal operation, three notches and four bends are required. If the operations can be done in any order, how many different ways of completing the manufacturing are possible? From equation 2 - 3, 7 3 4 35 ! ! !  sequences are possible 2.2.9 . In the laboratory analysis of samples from a chemical process, five samples from the process are analyzed daily. In addition, a control sample is analyzed twice each day to check the calibration of the laboratory instruments. (a) How many different sequences of process and control samples are possible each day? Assume that the five process samples are considered identical and that the two control samples are considered identical.

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 11 (b) How many different sequences of process and control samples are possible if we consider the five process samples to be different and the two control samples to be identical? (c) For the same situation as part (b), how many sequences are possible if the first test of each day must be a control sample? ( a) 21 ! 5 ! 2 ! 7  sequences are possible. ( b) 2520 ! 2 ! 1 ! 1 ! 1 ! 1 ! 1 ! 7  sequences are possible. ( c) 6! = 720 sequences are possible. 2.2.10 . In the layout of a printed circuit board for an electronic product, 12 different locations can accommodate chips. (a) If five different types of chips are to be placed on the board, how many different layouts are possible? (b) If the five chips that are placed on the board are of the same type, how many different layouts are possible? ( a) If the chips are of different types, then every arrangement of 5 locations selected from the 12 results in a different layout. Therefore, 040 , 95 ! 7 ! 12 12 5   P layouts are possible. ( b) If the chips are of the same type, then every subset of 5 locations chosen from the 12 results in a different layout. Therefore,   792 ! 7 ! 5 ! 12 12 5   layouts are possible. 2.2.11 . Consider the design of a communication system. (a) How many three - digit phone prefixes that are used to represent a particular geographic area (such as an area code) can be created from the digits 0 through 9? (b) As in part (a), how many three - digit phone prefixes are possible that do not start with 0 or 1, but contain 0 or 1 as the middle digit? (c) How many three - digit phone prefixes are possible in which no digit appears more than once in each prefix? ( a) From the multiplication rule, 10 3 1000  prefixes are possible ( b) From the multiplication rule, 8 2 10 160    are possible ( c) Every arrangement of three digits selected from the 10 digits results in a possible prefix. P 3 10 10 7 720   ! ! prefixes are possible. 2.2.12 . In the design of an electromechanical product, 12 components are to be stacked into a cylindrical casing in a manner that minimizes the impact of shocks. One end of the casing is designated as the bottom and the other end is the top. (a) If all components are different, how many different designs are possible? (b) If seven components are identical to one another, but the others are different, how many different designs are possible? (c) If three components are of one type and identical to one another, and four components are of another type and identical to one another, but the others are different, how many different designs are possible? ( a) Every arrangement selected from the 12 different components comprises a different design. Therefore, 600 , 001 , 479 ! 12  designs are possible. ( b) 7 components are the same, others are different, 95040 ! 1 ! 1 ! 1 ! 1 ! 1 ! 7 ! 12  designs are possible. ( c) 3326400 ! 4 ! 3 ! 12  designs are p ossible .

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 12 2.2.13 . A bin of 50 parts contains 5 that are defective. A sample of 10 parts is selected at random, without replacement. How many samples contain at least four defective parts? From the 5 defective parts, select 4, and the number of ways to complete this step is 5!/(4!1!) = 5 From the 45 non - defective parts, select 6, and the number of ways to complete this step is 45!/(6!39!) = 8,145,060 Therefore, the number of samples that contain exactly 4 defective parts is 5(8,145,060) = 40,725,300 Similarly, from the 5 defective parts, the number of ways to select 5 is 5!(5!1!) = 1 From the 45 non - defective parts, select 5, and the number of ways to complete this step is 45!/(5!40!) = 1,221,759 Therefore, the number of samples that contain exactly 5 defective parts is 1(1,221,759) = 1,221,759 Therefore, the number of samples that contain at least 4 defective parts is 40,725,300 + 1,221,759 = 41,947,059 2.2.14 . Plastic parts produced by an injection - molding operation are checked for conformance to specifications. Each tool contains 12 cavities in which parts are produced, and these parts fall into a conveyor when the press opens. An inspector chooses 3 parts from among the 12 at random. Two cavities are affected by a temperature malfunction that results in parts that do not conform to specifications. (a) How many samples contain exactly 1 nonconforming part? (b) How many samples contain at least 1 nonconforming part? ( a) The total number of samples is   3 12 12 3 9 220   ! ! ! . The number of samples that result in one nonconforming part is    . 90 ! 8 ! 2 ! 10 ! 1 ! 1 ! 2 10 2 2 1    Therefore, the requested probability is 90/220 = 0.409. ( b) The number of samples with no nonconforming part is   . 120 ! 7 ! 3 ! 10 10 3   The probability of at least one nonconforming part is 1 455 . 0 220 120   . 2.2.15 . A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. S uppose that an operating room needs to handle three knee, four hip, and five shoulder surgeries. (a) How many different sequences are possible? (b) How many different sequences have all hip, knee, and shoulder surgeries scheduled consecutively? (c) How many different schedules begin and end with a knee surgery? (a) From the formula for the number of sequences 12 ! 3 ! 4 ! ହ ! = 27,720 sequences are possible. (b) Combining all hip surgeries into one single unit, all knee surgeries into one single unit and all shoulder surgeries into one unit, the possible number of sequences of these units = 3! = 6 (c)With two surgeries specified, 10 remain and there are 10 ! 4 ! ହ ! 1 ! = 1,260 different sequences. Section 2 - 3 2.3.1 . The sample space of a random experiment is { a , b , c , d , e } with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively. Let A denote the event { a , b , c }, and let B denote the event { c , d , e }. Determine the following: (a) P(A) (b) P(B) (c) P(A') (d) P(A  B) (e) P(A  B ) ( a) P(A) = 0.4 ( b) P(B) = 0.8 ( c) P(A') = 0.6 ( d) P(A  B) = 1

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 13 ( e) P(A  B) = 0.2 2.3.2 . A part selected for testing is equally likely to have been produced on any one of six cutting tools. (a) What is the sample space? (b) What is the probability that the part is from tool 1? (c) What is the probability that the part is from tool 3 or tool 5? (d) What is the probability that the part is not from tool 4? ( a) S = {1, 2, 3, 4, 5, 6} ( b) 1/6 ( c) 2/6 ( d) 5/6 2.3.3 . An injection - molded part is equally likely to be obtained from any one of the eight cavities on a mold. (a) What is the sample space? (b) What is the probability that a part is from cavity 1 or 2? (c) What is the probability that a part is from neither cavity 3 nor 4? ( a) S = {1,2,3,4,5,6,7,8} ( b) 2/8 ( c) 6/8 2.3.4. A credit card contains 16 digits between 0 and 9. However, only 100 million numbers are valid. If a number is entered randomly, what is the probability that it is a valid number? Total possible: 10 16 , but o nly 10 8 are valid . Therefore, P(valid) = 10 8 /10 16 = 1/10 8 2.3.5 . In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states and that is usually found in the following states: (a) What is the probability that a cell has at least one of the positive nickel - charged options? (b) What is the probability that a cell is not composed of a positive nickel charge greater than +3? The sample space is {0, +2, +3, and +4}. (a) The event that a cell has at least one of the positive nickel charged options is {+2, +3, and +4}. The probability is 0.35 + 0.33 + 0.15 = 0.83. (b) The event that a cell is not composed of a positive nickel charge greater than +3 is {0, +2, and +3}. The probability is 0.17 + 0.35 + 0.33 = 0.85. 2.3.6. A message can follow different paths through servers on a network. The sender’s message can go to one of five servers for the first step; each of them can send to five servers at the second step; each of those can send to four servers at the third step; and then the message goes to the recipient’s server. (a) How many paths are possible? (b) If all paths are equally likely, what is the probability that a message passes through the first of four servers at the third step? ( a) 5*5*4 = 100 ( b) (5*5)/100 = 25/100=1/4

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 14 2.3.7 . Suppose your vehicle is licensed in a state that issues license plates that consist of three digits (between 0 and 9) followed by three letters (between A and Z ). If a license number is selected randomly, what is the probability that yours is the one selected? 3 digits between 0 and 9, so the probability of an y three numbers is 1/(10*10*10). 3 letters A to Z, so the probability of any three numbers is 1/(26*26*26) . The probability your license plate is chosen is then (1/10 3 )*(1/26 3 ) = 5.7 x 10 - 8 2.3.8. Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. If a disk is selected at random, determine the following probabilities: (a) P(A) (b) P(B) (c) P(A') (d) P(A  B) (e) P(A  B) (f) P(A’  B) ( a) P(A) = 86/100 = 0.86 ( b) P(B) = 79/100 = 0.79 ( c) P(A') = 14/100 = 0.14 ( d) P(A  B) = 70/100 = 0.70 ( e) P(A  B) = (70+9+16)/100 = 0.95 ( f) P(A’  B) = (70+9+5)/100 = 0.84 2.3.9 . Magnesium alkyls are used as homogenous catalysts in the production of linear low - density polyethylene (LLDPE), which requires a finer magnesium powder to sustain a reaction. Redox reaction experiments using four different amounts of magnesium powder are performed. Each result may or may not be further reduced in a second step using three different magnesium powder amounts. Each of these results may or may not be further reduced in a third step using three different amounts of magnesium powder. (a) How many experiments are possible? (b) If all outcomes are equally likely, what is the probability that the best result is obtained from an experiment that uses all three steps? (c) Does the result in part (b) change if five or six or seven different amounts are used in the first step? Explain. (a) The number of possible experiments is 4 + 4 × 3 + 4 × 3 × 3 = 52 (b) There are 36 experiments that use all three steps. The probability the best result uses all three steps is 36/52 = 0.6923. (c) No, it will not change. With k amounts in the first step the number of experiments is k + 3 k + 9 k = 13 k . The number of experiments that complete all three steps is 9 k out of 13 k . The probability is 9/13 = 0.6923. 2.3.10 . An article in the Journal of Database Management [“Experimental Study of a Self - Tuning Algorithm for DBMS Buffer Pools” (2005, Vol. 16, pp. 1 – 20)] provided the workload used in the TPC - C OLTP (Transaction Processing Performance Council’s Version C On - Line Transaction Processing) benchmark, which simulates a typical order entry application.

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    Applied Statistics and Probability for Engineers, 7 th edition 2017 2 - 15 The frequency of each type of transaction (in the second column) can be used as the percentage of each type of transaction. The average number of selects operations required for each type of transaction is shown. Let A denote the event of transactions with an average number of selects operations of 12 or fewer. Let B denote the event of transactions with an average number of updates operations of 12 or fewer. Calculate the following probabilities. (a) P(A) (b) P(B) (c) P(A  B) (d) P(A  B ’ ) (f) P(A  B) (a) The total number of transactions is 43+44+4+5+4=100 52 . 0 100 4 4 44 ) (     A P (b) 95 . 0 100 5 100 ) (    B P (c) 52 . 0 100 4 4 44 ) (      B A P (d) 0 ) ' (   B A P (e) 95 . 0 100 5 100 ) (     B A P 2.3.11 . Samples of emissions from three suppliers are classified for conformance to air - quality specifications. The results from 100 samples are summarized as follows: Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications. If a sample is selected at random, determine the following probabilities: (a) P(A) (b) P(B) (c) P(A') (d) P(A  B) (e) P(A  B) (f) P(A’  B) ( a) P(A) = 30/100 = 0.30 ( b) P(B) = 77/100 = 0.77 ( c) P(A') = 1 – 0.30 = 0.70 ( d) P(A  B) = 22/100 = 0.22 ( e) P(A  B) = 85/100 = 0.85 ( f) P(A’  B) =92/100 = 0.92 2.3.12. Consider the hospital emergency room data is given below. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any hospital).

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