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Miller And Freunds Probability And Statistics For Engineers 9th Edition Richard A. Johnson - Solutions

4.13 Use Table 1, or software, to find (a) B(7; 18, 0.45); (b) b(7; 18, 0.45); (c) B(8; 11, 0.95); (d) b(8; 11, 0.95);(e) 11 k=4 b( k; 11, 0.35); (f) 4 k=2 b( k; 10, 0.30).
4.12 Use Table 1, or software, to find (a) B(8; 16, 0.40); (b) b(8; 16, 0.40); (c) B(9; 12, 0.60); (d) b(9; 12, 0.60); (e) 20 k=6 b( k; 20, 0.15); (f) 9 k=6 b( k; 9, 0.70).
4.11 Which conditions for the binomial distribution, if any, fail to hold in the following situations? (a) The number of persons having a cold at a family reunion attended by 30 persons. (b) Among 8 projectors in the department office, 2 do not work properly but are not marked defective. Two are
4.10 What conditions for the binomial distribution, if any, fail to hold in the following situations? (a) For each of a company’s eight production facilities, record whether or not there was an accident in the past week. The largest facility has three times the number of production workers as
4.9 Do the assumptions for Bernoulli trials appear to hold? Explain. If the assumptions hold, identify success and the probability of interest. (a) A TV ratings company will use their electronic equipment to check a sample of homes around the city to see whether or not each has a set tuned to the
4.8 Prove that B( x; n, p) = 1 − B( n − x − 1; n, 1 − p).
4.7 Prove that b( x; n, p) = b( n − x; n, 1 − p).
4.6 With reference to Exercise 4.5, find an expression for the distribution function F(x) of the random variable.
4.5 Given that f (x) = k 2x is a probability distribution for a random variable that can take on the values x = 0, 1, 2, 3, and 4, find k.
4.4 Check whether the following can define probability distributions and explain your answers. (a) f (x) = 1 4 for x = 10, 11, 12, 13 (b) f (x) = 2x 5 for x = 0, 1, 2, 3, 4, 5 (c) f (x) = x − 15 20 for x = 8, 9, 10, 11, 12 (d) f (x) = 1 + x2 61 for x = 0, 1, 2, 3, 4, 5
4.3 Determine whether the following can be probability distributions of a random variable which can take on only the values 1, 2, 3, and 4. (a) f (1) = 0.19, f (2) = 0.27, f (3) = 0.27, and f (4) = 0.27; (b) f (1) = 0.24, f (2) = 0.24, f (3) = 0.24, and f (4) = 0.24; (c) f (1) = 0.35, f (2) = 0.33,
4.2 An experiment consists of five draws from a pack of cards. Denoting the outcomes BRRBR, BRRRB,..., for black and red cards and assuming that all 32 outcomes are equally likely, find the probability for the total number of red cards.
4.1 Suppose that a probability of 1 16 is assigned to each point of the sample space of part (a) of Exercise 3.1 on page 65. Find the probability distribution of the total number of units of black and white cement that are adulterated.
3.100 Long run relative frequency interpretation of probability. A simulation. A long series of experiments can be simulated using MINITAB and then the relative frequencies plotted as in Figure 3.7b.
3.99 Amy commutes to work by two different routes A and B. If she comes home by route A, then she will be home no later than 6 p.m. with probability 0.8, but if she comes home by route B, then she will be home no later than 6 p.m. with probability 0.7. In the past, the proportion of times that Amy
3.98 During the inspection of a rejected integrated circuit (IC), it was observed that the rejected IC could have an incorrect circuit, it could be bent, or it could have both defects. The probability of having an incorrect circuit is 0.45, the probability of being bent is 0.65, and the probability
3.97 An explosion in an LNG storage tank in the process of being repaired could have occurred as the result of static electricity, malfunctioning electrical equipment, an open flame in contact with the liner, or purposeful action (industrial sabotage). Interviews with engineers who were analyzing
3.96 Refer to Exercise 3.95. Given that a student, selected at random, is found to have an extensive understanding of physics, what is the probability that the student has (a) an extensive understanding of chemistry; (b) an extensive understanding of both chemistry and mathematics; (c) an extensive
3.95 The following frequency table shows the classification of 90 students in their sophomore year of college according to their understanding of physics, chemistry, and mathematics. Physics Average Extensive Chemistry Chemistry Average Extensive Average Extensive Mathematics Average 8 16 12 18
3.94 If events A and B are independent, and P(A) = 0.45 and P(B) = 0.20, find (a) P(A ∩ B); (b) P(A ∪ B); (c) P(A ∪ B ); (d) P(B | A).
3.93 Given P(A) = 0.40, P(B) = 0.55 and P(A ∩ B) = 0.10, find (a) P(A | B ); (b) P(A | B ); (c) P(B | A ); (d) P(B | A ).
3.92 The probabilities that a satellite launching rocket will explode during lift-off or have its guidance system fail in flight are 0.0002 and 0.0005. Assuming independence find the probabilities that such a rocket will (a) not explode during lift-off; (b) explode during lift-off or have its
3.91 The marketing manager reported to the head engineer regarding a survey concerning the company’s portable cleaning tool. He claims that, among the 200 customers surveyed, 165 said the product is reliable, 117 said it is easy to use, 88 said it is both reliable and easy to use, and 33 said it
3.90 In a sample of 652 engines tested, only 28 of them have cylinders with a mild leak. Estimate the probability that an engine tested will have a leak in its cylinder.
3.89 Given P(A) = 0.30, P(B) = 0.40, and P(A ∩ B) = 0.20, find (a) P(A ∪ B); (b) P( A ∩ B); (c) P(A ∩ B ); (d) P( A ∪ B ). (e) Are A and B independent?
3.88 Refer to Example 12 of motors for miniaturized capsules, but instead suppose that 20 motors are available and that 4 will not operate satisfactorily, when placed in a capsule. If the scientist wishes to fabricate two capsules, with two motors each, find the probability that among the four
3.87 In how many ways can 4 out of 11 similar propellers be fitted on an airplane?
3.86 CCTV cameras are to be fitted at six fixed places in a bank. In how many ways can the six available CCTV cameras be fitted at the six places in the bank?
3.85 The quality of surround sound from four digital movie systems is to be rated superior, average, or inferior, and we are interested only in how many of the systems get each of these ratings. Draw a tree diagram which shows the 12 different possibilities.
3.84 Use Venn diagrams to verify that (a) A ∩ B = A ∪ B ; (b) A ∪ B = A ∩ B .
3.83 With reference to the preceding exercise, express each of the following events symbolically by listing its elements, and also express it in words: (a) Y; (b) X ∩ Y; (c) Y ∪ Z; (d) X ∪ Y.
3.82 A construction engineer has to inspect 5 construction sites in a 2-day inspection schedule. He may or may not be able to visit these sites in two days. He will not visit any site more than once. (a) Using two coordinates so that (3, 1), for example, represents the event that he will visit 3
3.81 (a) Last year, 425 companies applied for 52 tenders floated by the government. This year, you will be applying for one of 52 similar tenders being floated, and would like to estimate the probability of being allotted one. Give your estimate and comment on one factor that might influence your
3.79 Engineers in charge of maintaining our nuclear fleet must continually check for corrosion inside the pipes that are part of the cooling systems. The inside condition of the pipes cannot be observed directly but a nondestructive test can give an indication of possible corrosion. This test is
3.78 Two firms V and W consider bidding on a roadbuilding job, which may or may not be awarded depending on the amounts of the bids. Firm V submits a bid and the probability is 0.8 that it will get the job provided firm W does not bid. The probability is 0.7 that W will bid, and if it does, the
3.77 With reference to the Example 30, for a problem diagnosed as being due to an incomplete initial repair, find the probability that the initial repair was made by (a) Tom; (b) Georgia; (c) Peter.
3.76 Refer to Example 31 concerning spam but now suppose that among the 5000 messages, the 1750 spam messages have 1570 that contain the words on a new list and that the 3250 normal messages have 300 that contain the words. (a) Find the probability that a message is spam given that the message
3.75 Refer to the example on page 84 but suppose the manufacturer has difficulty getting enough LED screens. Because of the shortage, the manufacturer had to obtain 40% of the screens from the second supplier and 15% from the third supplier. Find the (a) probability that a LED screen will meet
3.74 Identity theft is a growing problem in the United States. According to a Federal Trade Commission Report about 280,000 identity complaints were filed for 2011. Among the 43.2 million persons in the 20–29 year old age group, 56,689 complaints were filed. The 20–29 year old age group makes
3.73 An insurance company’s records of 12,299 automobile insurance policies showed that 2073 policy holders made a claim (Courtesy J. Hickman). Among insured drivers under age 25, there were 1032 claims out of 5192 policies. For person selected at random from the policy holders, let A = [Claim
3.72 There are over twenty thousand objects orbiting in space. For a given object, let A be the event that the charred remains do hit the earth. Suppose experts, using their knowledge of the size and composition of the object as well as its re-entry angle, determine that P(A) = 0.25. Next, let B
3.71 Use the information on the tree diagram of Figure 3.17 to determine the value of (a) P(Y ); (b) P(X |Y ); (c) P(X |Y ). 0.45 x 0.35 Y Y X 0.75 8 Y
3.70 For three or more events which are not independent, the probability that they will all occur is obtained by multiplying the probability that one of the events will occur, times the probability that a second of the events will occur given that the first event has occurred, times the
3.69 Find the probabilities of getting (a) eight heads in a row with a balanced coin; (b) three 3’s and then a 4 or a 5 in four rolls of a balanced die; (c) five multiple-choice questions answered correctly, if for each question the probability of answering it correctly is 1 3 .
3.68 If the odds are 5 to 3 that an event M will not occur, 2 to 1 that event N will occur, and 4 to 1 that they will not both occur, are the two events M and N independent?
3.67 If P(X) = 0.33, P(Y ) = 0.75, and P(X ∩ Y ) = 0.30, are X and Y independent?
3.66 A large firm has 85% of its service calls made by a contractor, and 10% of these calls result in customer complaints. The other 15% of the service calls are made by their own employees, and these calls have a 5% complaint rate. Find the (a) probability of receiving a complaint. (b)
3.63 Given that P (A) = 0.60, P (B) = 0.40, and P (A∩B) = 0.24, verify that (a) P(A | B) = P(A); (b) P(A | B ) = P(A);(c) P(B | A) = P(B); (d) P(B | A ) = P(B). 3.64 Among 40 condensers produced by a machine, 6 are defective. If we randomly check 5 condensers, what are the probabilities that (a)
3.62 In a certain city, sports bikes are being targeted by thieves. Assume that the probability of a sports bike being stolen is 0.09 while the probability is only 0.5 for a regular bike. Taking, as an approximation for all bikes in that area, the nationwide proportion 0.19 of sports bikes, find
3.61 Prove that P(A | B) = P(A) implies that P(B | A) = P(B) provided that P(A) = 0 and P(B) = 0.
3.60 With reference to Exercise 3.47, find the probabilities that the company will get the tender for constructing an underpass given that (a) it got the tender for constructing a flyover; (b) it did not get the tender for constructing a flyover.
3.59 With reference to the used car example and the probabilities given in Figure 3.9, find (a) P(M1 | P1) and compare its value with that of P(M1); (b) P(C3 | P2) and compare its value with that of P(C3); (c) P(M1 | P1 ∩C3) and compare its value with that of P(M1).
3.58 With reference to Figure 3.13, find (a) P(A | B); (b) P(B |C ); (c) P(A ∩ B |C); (d) P(B ∪ C | A ); (e) P(A | B ∪ C); (f) P(A | B ∩ C); (g) P(A ∩ B ∩ C | B ∩ C); (h) P(A ∩ B ∩ C | B ∪ C).
3.57 With reference to Exercise 3.34 and Figure 3.11, assume that each of 150 persons has the same chance of being selected, and find the probabilities that he or she (a) lives more than 3 miles from the center of the city given that he or she would gladly switch to public mass transportation; (b)
3.56 (a) Would you expect the probability that a randomly selected car will need major repairs in the next year to be smaller, remain the same, or increase if you are told it already has high mileage? Explain. (b) Would you expect the probability that a randomly selected senior would know the
3.55 With reference to Figure 3.8, find P(I | D) and P(I | D ), assuming that originally each of the 500 machine parts has the same chance of being chosen for inspection.
3.54 Subjective probabilities may or may not satisfy the third axiom of probability. When they do, we say that they are consistent; when they do not, they ought not to be taken too seriously. (a) The supplier of delicate optical equipment feels that the odds are 7 to 5 against a shipment arriving
3.53 The formula of Exercise 3.52 is often used to determine subjective probabilities. For instance, if an applicant for a job “feels” that the odds are 7 to 4 of getting the job, the subjective probability the applicant assigns to getting the job is p = 7 7 + 4 = 7 11 (a) If a businessperson
3.52 Use the definition of Exercise 3.51 to show that if the odds for the occurrence of event A are a tob, where a and b are positive integers, then p = a a + b
3.51 If the probability of event A is p, then the odds that it will occur are given by the ratio of p to 1− p. Odds are usually given as a ratio of two positive integers having no common factor, and if an event is more likely not to occur than to occur, it is customary to give the odds that it
3.50 Suppose that in the maintenance of a large medicalrecords file for insurance purposes the probability of an error in processing is 0.0010, the probability of an error in filing is 0.0009, the probability of an error in retrieving is 0.0012, the probability of an error in processing as well
3.49 It can be shown that for any three events A, B, and C, the probability that at least one of them will occur is given by P( A ∪ B ∪ C ) = P( A ) + P( B ) + P(C ) − P( A ∩ B ) − P( A ∩ C ) − P( B ∩ C ) + P( A ∩ B ∩ C ) Verify that this formula holds for the probabilities of
3.48 Given P(A) = 0.30, P(B) = 0.62, and P( A ∩ B ) = 0.12, find (a) P( A ∪ B ); (b) P( A ∩ B ); (c) P(A ∩ B ); (d) P( A ∪ B )
3.47 The probability that a construction company will get the tender for constructing a flyover is 0.33, the probability that it will get the tender for constructing an underpass is 0.28, and the probability that it will get both tenders is 0.13. (a) What is the probability that it will get at
3.46 The probability that a turbine will have a defective coil is 0.10, the probability that it will have defective blades is 0.15, and the probability that it will have both defects is 0.04. (a) What is the probability that a turbine will have one of these defects? (b) What is the probability that
3.45 If each point of the sample space of Figure 3.12 represents an outcome having the probability 1 32 , find (a) P(A); (b) P(B); (c) P(A ∩ B ); (d) P(A ∪ B ); (e) P( A ∩ B ); (f) P( A ∩ B ). A B S
3.44 The probabilities that a TV station will receive 0, 1, 2, 3,..., 8 or at least 9 complaints after showing a controversial program are, respectively,0.01, 0.03, 0.07, 0.15, 0.19, 0.18, 0.14, 0.12, 0.09, and 0.02. What are the probabilities that after showing such a program the station will
3.43 A rotary plug valve needs to be replaced to repair a machine, and the probabilities that the replacement will be a flange style (low pressure), flange style (high pressure),wafer style, or lug style are 0.16, 0.29, 0.26, and 0.15. Find the probabilities that the replacement will be (a) a
3.42 With reference to Exercise 3.34, suppose that the questionnaire filled in by one of the 150 persons is to be double-checked. If it is chosen in such a way that each questionnaire has a probability of 1 150 of being selected, find the probabilities that the person (a) lives more than 3 miles
3.41 If A and B are mutually exclusive events, P(A) = 0.45, and P(B) = 0.30, find (a) P( A ); (b) P( A ∪ B ); (c) P( A ∩ B ); (d) P( A ∩ B ).
3.40 Explain why there must be a mistake in each of the following statements: (a) The probability that a student will get an A in a geology course is 0.3, and the probability that he or she will get either an A or a B is 0.27. (b) A company is working on the construction of two shopping centers;
3.39 Refer to parts (d) and (c) of Exercise 3.13 to show that (a) P(A ∩ B) ≤ P(A); (b) P(A ∪ B) ≥ P(A).
3.38 Explain why there must be a mistake in each of the following statements: (a) The probability that a mineral sample will contain silver is 0.38 and the probability that it will not contain silver is 0.52. (b) The probability that a drilling operation will be a success is 0.34 and the
3.37 With reference to Exercise 3.7, suppose that each point (i, j) of the sample space is assigned the probability 420/401 2(i + j) . (a) Verify that this assignment of probabilities is permissible. (b) Find the probabilities of events X, Y, and Z described in part (b) of that exercise. (c) Find
3.36 With reference to Exercise 3.1, suppose that the points (0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 1), (2, 2), (2, 3), (3, 0), (3, 1), (3, 2), and (3, 3) have the probabilities 0.080, 0.032, 0.086, 0.064, 0.085, 0.073, 0.065, 0.091, 0.071, 0.050, 0.046, 0.075,
3.35 An experiment has the four possible mutually exclusive outcomes A, B,C, and D. Check whether the following assignments of probability are permissible: (a) P(A) = 0.38, P(B) = 0.16, P(C) = 0.11, P(D) = 0.35; (b) P(A) = 0.27, P(B) = 0.30, P(C) = 0.28, P(D) = 0.16; (c) P(A) = 0.32, P(B) = 0.27,
3.34 Among 150 persons interviewed as part of an urban mass transportation study, some live more than 3 miles from the center of the city (A), some now regularly drive their own car to work (B), and some would gladly switch to public mass transportation if it were available (C). Use the information
course?
3.33 In a group of 160 graduate engineering students, 92 are enrolled in an advanced course in statistics, 63 are enrolled in a course in operations research, and 40 are enrolled in both. How many of these students are not enrolled in either
3.32 Last year; the maximum daily temperature in a plants’ server room exceeded 68◦F in 12 days. Estimate the probability that the maximum temperature will exceed 68◦F tomorrow.
3.31 A car rental agency has 19 compact cars and 12 intermediate-size cars. If four of the cars are randomly selected for a safety check, what is the probability of getting two of each kind?
3.30 The registration numbers for the candidates of an entrance test are numbered from 000001 to 200000. What is the probability that a candidate will get a registration number divisible by 40?
3.29 When we roll a pair of balanced dice, what are the probabilities of getting (a) 7; (b) 11; (c) 7 or 11; (d) 3; (e) 2 or 12; (f) 2, 3, or 12?
3.28 (a) Among 880 smart phones sold by a retailer, 72 required repairs under the warranty. Estimate the probability that a new phone, which has just been sold, will require repairs under the warranty. Explain your reasoning. (b) Last year 8,400 students applied for the 6,000 student season
3.27 An engineering student has 6 different ball bearings and 9 different gears. In how many ways can 3 ball bearings and 3 gears be selected for an experiment on friction in machine parts?
3.26 With reference to Exercise 3.25, suppose that three of the spark plugs are defective. In how many ways can 4 spark plugs be selected so that (a) one of the defective plugs is selected; (b) two of the defective plugs are selected; (c) all three defective plugs are selected?
3.25 A box of 15 spark plugs contains one that is defective. In how many ways can 4 spark plugs be selected so that (a) the defective one is selected; (b) the defective plug is not selected?
3.24 How many ways can a company select 4 candidates to interview from a short list of 12 engineers?
3.23 Determine the number of ways in which a software professional can choose 4 of 25 laptops to test a newly designed application.
3.22 If among n objects k are alike and the others are all distinct, the number of permutations of these n objects taken all together is n!/k!. (a) How many permutations are there of the letters of the word class? (b) In how many ways can the television director of Exercise 3.21 fill the 6 time
3.21 In how many ordered ways can a television director schedule 6 different commercials during the 6 time slots allocated to commercials during the telecast of the first period of a hockey game?
3.20 If there are 9 cars in a race, in how many different ways can they place first, second, and third?
3.19 An Engineers Association consists of 5 civil engineers and 5 mechanical engineers. (a) In how many ways can a committee of 3 civil engineers and 2 mechanical engineers be appointed? (b) If 2 civil engineers disagree with each other and refuse to be on the same committee together, how many
3.18 You are required to choose a four digit personal identification number (PIN) for a new debit card. Each digit is selected from 0, 1,..., 9. How many choices do you have.
3.17 Students are offered three cooperative training programs at local companies and four training programs outside the state. Count the number of possible training opportunities if an opportunity consists of training at (a) one local company or one company outside of the state.(b) one local
3.16 If a number cannot be immediately repeated, how many different three number combinations are possible for a combination lock with numbers 0, 1,..., 29.
3.15 If the five finalists in an international volleyball tournament are Spain, the United States, Uruguay, Portugal, and Japan, draw a tree diagram that shows the various possible first- and second-place finishers.
3.14 A building inspector has to check the wiring in a new apartment building either on Monday, Tuesday, Wednesday, or Thursday, and at 8 a.m., 1 p.m., or 2 p.m. Draw a tree diagram which shows the various ways in which the inspector can schedule the inspection of the wiring of the new apartment
3.13 Use Venn diagrams to verify that (a) A ∪ B = A ∩ B (b) B ∩ (A ∪ B) = B (c) (A ∪ B) ∩ (A ∪ B) = B (d) A ∩ B = (A ∪ B) ∩ (A ∪ B ) ∩ (A ∪ B ) (e) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
3.12 With reference to Figure 3.4, what regions or combinations of regions represent the events that a motor will have (a) none of the major defects; (b) a shaft that is large and windings improper; (c) a shaft that is large and/or windings improper but the electrical connections are satisfactory;

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