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applied statistics and probability for engineers

Probability And Statistics For Engineering And The Sciences 7th Edition Dave Ellis, Jay L Devore - Solutions

c. Given that at least one of the two projects is successful, what is the probability that only the Asian project is successful?
b. What is the probability that at least one of the two projects will be successful?
a. If the Asian project is not successful, what is the probability that the European project is also not successful?Explain your reasoning.
71. An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian project is successful and B be the event that the European project is successful. Suppose that A and B are independent events with P(A) .4 and P(B) .7.
70. Reconsider the credit card scenario of Exercise 47 (Section 2.4), and show that A and B are dependent first by using the definition of independence and then by verifying that the multiplication property does not hold.
69. In Exercise 59, consider the following additional information on credit card usage:70% of all regular fill-up customers use a credit card.50% of all regular non-fill-up customers use a credit card.60% of all plus fill-up customers use a credit card.50% of all plus non-fill-up customers use a
68. A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; 50% of the time she travels on airline #1, 30% of the time on airline #2, and the remaining 20% of the time on airline #3. For airline #1, flights are late into D.C. 30% of the time and late into L.A. 10% of
67. There has been a great deal of controversy over the last several years regarding what types of surveillance are appropriate to prevent terrorism. Suppose a particular surveillance system has a 99% chance of correctly identifying a future terrorist and a 99.9% chance of correctly identifying
64. In Example 2.30, suppose that the incidence rate for the disease is 1 in 25 rather than 1 in 1000. What then is the probability of a positive test result? Given that the test result is positive, what is the probability that the individual has the disease? Given a negative test result, what is
63. For customers purchasing a refrigerator at a certain appliance store, let A be the event that the refrigerator was manufactured in the U.S., B be the event that the refrigerator had an icemaker, and C be the event that the customer purchased an extended warranty. Relevant probabilities are.P(A)
62. A company that manufactures video cameras produces a basic model and a deluxe model. Over the past year, 40% of the cameras sold have been of the basic model. Of those buying the basic model, 30% purchase an extended warranty, whereas 50% of all deluxe purchasers do so. If you learn that a
61. Components of a certain type are shipped to a supplier in batches of ten. Suppose that 50% of all such batches contain no defective components, 30% contain one defective component, and 20% contain two defective components. Two components from a batch are randomly selected and tested.What are
60. Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose a light aircraft has
59. At a certain gas station, 40% of the customers use regular gas (A1), 35% use plus gas (A2), and 25% use premium (A3).Of those customers using regular gas, only 30% fill their tanks (event B). Of those customers using plus, 60% fill their tanks, whereas of those using premium, 50% fill their
58. Show that for any three events A, B, and C with P(C) 0, P(A B⏐C) P(A⏐C) P(B⏐C) P(A B⏐C).
57. If P(B⏐A) P(B), show that P(B ⏐A) P(B ). [Hint: Add P(B ⏐A) to both sides of the given inequality and then use the result of Exercise 56.]
56. For any events A and B with P(B) 0, show that P(A⏐B)P(A ⏐B) 1.
55. Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a recent study, suppose that 16% of all ticks in a certain location carry Lyme disease, 10% carry HGE, and 10% of the ticks that carry at least one of these diseases in fact carry both of them.
54. In Exercise 13, Ai {awarded project i}, for i 1, 2, 3.Use the probabilities given there to compute the following probabilities, and explain in words the meaning of each one.a. P(A2⏐A1)b. P(A2 A3⏐A1)c. P(A2 A3⏐A1)d. P(A1 A2 A3⏐A1 A2 A3).
53. A certain shop repairs both audio and video components.Let A denote the event that the next component brought in for repair is an audio component, and let B be the event that the next component is a compact disc player (so the event B is contained in A). Suppose that P(A) .6 and P(B)
52. A system consists of two identical pumps, #1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the extra remaining pump is now more likely to fail than was originally the case. That is, r P(#2 fails ⏐#1 fails) P(#2 fails) q. If at least one
51. One box contains six red balls and four green balls, and a second box contains seven red balls and three green balls. A ball is randomly chosen from the first box and placed in the second box. Then a ball is randomly selected from the second box and placed in the first box.a. What is the
f. Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleeved?
e. Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?
d. What is the probability that the size of the next shirt sold is medium? That the pattern of the next shirt sold is a print?
c. What is the probability that the next shirt sold is a shortsleeved shirt? A long-sleeved shirt?
b. What is the probability that the next shirt sold is a medium print shirt?
a. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?
50. A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations.Short-sleeved Pattern Size Pl Pr
49. If two bulbs are randomly selected from the box of lightbulbs described in Exercise 38 (Section 2.3) and at least one of them is found to be rated 75 W, what is the probability that both of them are 75-W bulbs? Given that at least one of the two selected is not rated 75 W, what is the
48. Reconsider the system defect situation described in Exercise 26 (Section 2.2).a. Given that the system has a type 1 defect, what is the probability that it has a type 2 defect?b. Given that the system has a type 1 defect, what is the probability that it has all three types of defects?c. Given
47. Return to the credit card scenario of Exercise 12 (Section 2.2), where A {Visa}, B {MasterCard}, P(A) .5, P(B) .4, and P(A B) .25. Calculate and interpret each of the following probabilities (a Venn diagram might help).a. P(B⏐A)b. P(B ⏐A)c. P(A⏐B)d. P(A ⏐B)e. Given that the
46. Suppose an individual is randomly selected from the population of all adult males living in the United States. Let A be the event that the selected individual is over 6 ft in height, and let B be the event that the selected individual is a professional basketball player. Which do you think is
45. The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group–blood group combinations.Suppose that an individual
44. Show that (n k) (nn k). Give an interpretation involving subsets.
What is the probability that it will be a straight?What is the probability that it will be a straight flush (all cards in the same suit)?
43. In five-card poker, a straight consists of five cards with adjacent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king of clubs). Assuming that aces can be high or low, if you are dealt a five-card hand, what is the probability that it will be a straight
42. Three married couples have purchased theater tickets and are seated in a row consisting of just six seats. If they take their seats in a completely random fashion (random order), what is the probability that Jim and Paula (husband and wife) sit in the two seats on the far left? What is the
c. Suppose the professor has the same eight assistants the following semester and again schedules appointments without regard to the ordering during the first semester. What is the probability that the orderings of appointments are different?
b. What is the probability that after the first five appointments she has met with all female assistants?
a. What is the probability that at least one female assistant is among the first three with whom the professor meets?
41. A mathematics professor wishes to schedule an appointment with each of her eight teaching assistants, four men and four women, to discuss her calculus course. Suppose all possible orderings of appointments are equally likely to be selected.
40. Three molecules of type A, three of type B, three of type C, and three of type D are to be linked together to form a chain molecule. One such chain molecule is ABCDABCDABCD, and another is BCDDAAABDBCC.a. How many such chain molecules are there? [Hint: If the three A’s were distinguishable
39. Fifteen telephones have just been received at an authorized service center. Five of these telephones are cellular, five are cordless, and the other five are corded phones. Suppose that these components are randomly allocated the numbers 1, 2, . . . , 15 to establish the order in which they will
38. A box in a certain supply room contains four 40-W lightbulbs, five 60-W bulbs, and six 75-W bulbs. Suppose that three bulbs are randomly selected.a. What is the probability that exactly two of the selected bulbs are rated 75 W?b. What is the probability that all three of the selected bulbs have
37. An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration.a. If any particular experimental run involves the use
36. An academic department with five faculty members narrowed its choice for department head to either candidate A or candidate B. Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for A and two for B. If the slips are selected for tallying
What is the probability that at least one of the shifts will be unrepresented in the sample of workers?
What is the probability that at least two different shifts will be represented among the selected workers?
35. A production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10 workers on the graveyard shift. A quality control consultant is to select 6 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 6
34. Shortly after being put into service, some buses manufactured by a certain company have developed cracks on the underside of the main frame. Suppose a particular city has 25 of these buses, and cracks have actually appeared in 8 of them.a. How many ways are there to select a sample of 5 buses
c. Suppose 5 of the 15 players are left-handed. How many ways are there to select 3 left-handed outfielders and have all 6 other positions occupied by right-handed players?
b. How many ways are there to select 9 players for the starting lineup and a batting order for the 9 starters?
a. How many ways are there to select 9 players for the starting lineup?
33. Again consider a Little League team that has 15 players on its roster.
e. If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component?Exactly one Sony component?
d. In how many ways can a selection be made if at least one Sony component is to be included?
c. In how many ways can components be selected if none is to be Sony?
a. In how many ways can one component of each type be selected?b. In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?
32. A stereo store is offering a special price on a complete set of components (receiver, compact disc player, speakers, cassette deck). A purchaser is offered a choice of manufacturer for each component:Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood Compact disc player: Onkyo, Pioneer, Sony,
31.a. Beethoven wrote 9 symphonies and Mozart wrote 27 piano concertos. If a university radio station announcer wishes to play first a Beethoven symphony and then a Mozart concerto, in how many ways can this be done?b. The station manager decides that on each successive night(7 days per week), a
30. A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries.a. If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to
29. As of April 2006, roughly 50 million .com web domain names were registered (e.g., yahoo.com).a. How many domain names consisting of just two letters in sequence can be formed? How many domain names of length two are there if digits as well as letters are permitted as characters? [Note: A
28. In Exercise 5, suppose that any incoming individual is equally likely to be assigned to any of the three stations irrespective of where other individuals have been assigned.What is the probability thata. All three family members are assigned to the same station?b. At most two family members are
27. An academic department with five faculty members—Anderson, Box, Cox, Cramer, and Fisher—must select two of its members to serve on a personnel review committee.Because the work will be time-consuming, no one is anxious to serve, so it is decided that the representative will be selected by
26. A certain system can experience three different types of defects. Let Ai (i 1, 2, 3) denote the event that the system has a defect of type i. Suppose that P(A1) .12 P(A2) .07 P(A3) .05 P(A1 A2) .13 P(A1 A3) .14 P(A2 A3) .10 P(A1 A2 A3) .01a. What is the probability that
25. The three major options on a certain type of new car are an automatic transmission (A), a sunroof (B), and a stereo with compact disc player (C). If 70% of all purchasers request A, 80% request B, 75% request C, 85% request A or B, 90%request A or C, 95% request B or C, and 98% request A or B
24. Show that if one event A is contained in another event B(i.e., A is a subset of B), then P(A) P(B). [Hint: For such A and B, A and B A are disjoint and B A (B A ), as can be seen from a Venn diagram.] For general A and B, what does this imply about the relationship among P(A B), P(A), and
23. The computers of six faculty members in a certain department are to be replaced. Two of the faculty members have selected laptop machines and the other four have chosen desktop machines. Suppose that only two of the setups can be done on a particular day, and the two computers to be set up are
22. The route used by a certain motorist in commuting to work contains two intersections with traffic signals. The probability
21. An insurance company offers four different deductible levels—none, low, medium, and high—for its homeowner’s policyholders and three different levels—low, medium, and high—for its automobile policyholders. The accompanying table gives proportions for the various categories of
20. A certain factory operates three different shifts. Over the last year, 200 accidents have occurred at the factory.Some of these can be attributed at least in part to unsafe working conditions, whereas the others are unrelated to working conditions. The accompanying table gives the percentage of
19. Human visual inspection of solder joints on printed circuit boards can be very subjective. Part of the problem stems from the numerous types of solder defects (e.g., pad nonwetting, knee visibility, voids) and even the degree to which a joint possesses one or more of these defects.
18. A box contains four 40-W bulbs, five 60-W bulbs, and six 75-W bulbs. If bulbs are selected one by one in random order, what is the probability that at least two bulbs must be selected to obtain one that is rated 75 W?
17. Let A denote the event that the next request for assistance from a statistical software consultant relates to the SPSS package, and let B be the event that the next request is for help with SAS. Suppose that P(A) .30 and P(B) .50.a. Why is it not the case that P(A) P(B) 1?b. Calculate P(A
16. An individual is presented with three different glasses of cola, labeled C, D, and P. He is asked to taste all three and then list them in order of preference. Suppose the same cola has actually been put into all three glasses.a. What are the simple events in this ranking experiment, and what
15. Consider the type of clothes dryer (gas or electric) purchased by each of five different customers at a certain store.a. If the probability that at most one of these purchases an electric dryer is .428, what is the probability that at least two purchase an electric dryer?b. If P(all five
14. A utility company offers a lifeline rate to any household whose electricity usage falls below 240 kWh during a
13. A computer consulting firm presently has bids out on three projects. Let Ai {awarded project i}, for i 1, 2, 3, and suppose that P(A1) .22, P(A2) .25, P(A3) .28, P(A1 A2) .11, P(A1 A3) .05, P(A2 A3) .07, P(A1 A2 A3) .01. Express in words each of the following events, and
12. Consider randomly selecting a student at a certain university, and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard.Suppose that P(A) .5, P(B) .4, and P(A B) .25.a. Compute the probability that the selected individual
11. A mutual fund company offers its customers several different funds: a money-market fund, three different bond funds (short, intermediate, and long-term), two stock funds (moderate and high-risk), and a balanced fund. Among customers who own shares in just one fund, the percentages of customers
10.a. In Example 2.10, identify three events that are mutually exclusive.b. Suppose there is no outcome common to all three of the events A, B, and C. Are these three events necessarily mutually exclusive? If your answer is yes, explain why;if your answer is no, give a counterexample using the
9. Use Venn diagrams to verify the following two relationships for any events A and B (these are called De Morgan’s laws):a. (A B) A Bb. (A B) A B
8. An engineering construction firm is currently working on power plants at three different sites. Let Ai denote the event that the plant at site i is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of
7. An academic department has just completed voting by secret ballot for a department head. The ballot box contains four slips with votes for candidate A and three slips with votes for candidate B. Suppose these slips are removed from the box one by one.a. List all possible outcomes.b. Suppose a
6. A college library has five copies of a certain text on reserve.Two copies (1 and 2) are first printings, and the other three(3, 4, and 5) are second printings. A student examines these books in random order, stopping only when a second printing has been selected. One possible outcome is 5, and
5. A family consisting of three persons—A, B, and C—belongs to a medical clinic that always has a doctor at each of stations 1, 2, and 3. During a certain week, each member of the family visits the clinic once and is assigned at random to a station. The experiment consists of recording the
4. Each of a sample of four home mortgages is classified as fixed rate (F) or variable rate (V).a. What are the 16 outcomes in S ?b. Which outcomes are in the event that exactly three of the selected mortgages are fixed rate?c. Which outcomes are in the event that all four mortgages are of the same
3. Three components are connected to form a system as shown in the accompanying diagram. Because the components in the 2–3 subsystem are connected in parallel, that subsystem will function if at least one of the two individual components functions. For the entire system to function, component 1
2. Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L), or go straight (S). Consider observing the direction for each of three successive vehicles.a. List all outcomes in the event A that all three vehicles go in the same direction.b. List all outcomes in the
1. Four universities—1, 2, 3, and 4—are participating in a holiday basketball tournament. In the first round, 1 will play 2 and 3 will play 4. Then the two winners will play for the championship, and the two losers will also play. One possible outcome can be denoted by 1324 (1 beats 2 and 3
83. Consider numerical observations x1, . . . , xn. It is frequently of interest to know whether the xis are (at least approximately)symmetrically distributed about some value. If n is at least moderately large, the extent of symmetry can be assessed from a stem-and-leaf display or histogram.
82. The sample data x1, x2, . . . , xn sometimes represents a time series, where xt the observed value of a response variable x at time t. Often the observed series shows a great deal of random variation, which makes it difficult to study longerterm behavior. In such situations, it is desirable
81. A study carried out to investigate the distribution of total braking time (reaction time plus accelerator-to-brake movement time, in ms) during real driving conditions at 60 km/hr gave the following summary information on the distribution of times (“A Field Study on Braking Responses during
80. Lengths of bus routes for any particular transit system will typically vary from one route to another. The article“Planning of City Bus Routes” (J. of the Institution of Engineers, 1995: 211–215) gives the following information on lengths (km) for one particular system:Length 6–8
79. Let and denote the sample mean and variance for the sample x1, . . . , xn and let and denote these quantities when an additional observation is added to the sample.a. Show how can be computed from and .b. Show that so that can be computed from , , and .c. Suppose that a sample of 15 strands of
78. Consider a sample x1, x2, . . . , xn and suppose that the values of , s2, and s have been calculated.a. Let yi xi for i 1, . . . , n. How do the values of s2 and s for the yis compare to the corresponding values for the xis? Explain.b. Let zi (xi )/s for i 1, . . . , n. What are the
77. Consider the following data on active repair time (hours) for a sample of n 46 airborne communications receivers:.2 .3 .5 .5 .5 .6 .6 .7 .7 .7 .8 .8.8 1.0 1.0 1.0 1.0 1.1 1.3 1.5 1.5 1.5 1.5 2.0 2.0 2.2 2.5 2.7 3.0 3.0 3.3 3.3 4.0 4.0 4.5 4.7 5.0 5.4 5.4 7.0 7.5 8.8 9.0 10.3 22.0 24.5
76. The three measures of center introduced in this chapter are the mean, median, and trimmed mean. Two additional measures of center that are occasionally used are the midrange, which is the average of the smallest and largest observations, and the midfourth, which is the average of the two
75. Specimens of three different types of rope wire were selected, and the fatigue limit (MPa) was determined for each specimen, resulting in the accompanying data.Type 1 350 350 350 358 370 370 370 371 371 372 372 384 391 391 392 Type 2 350 354 359 363 365 368 369 371 373 374 376 380 383 388 392

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