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.
> The article “Clinical and Radiographic Outcomes of Four Different Treatment Strategies in Patients with Early Rheumatoid Arthritis” [Arthritis & Rheumatism (2005, Vol. 52, pp. 3381–3390)] considered four treatment groups. The groups consisted of patients
> Consider the three patient groups in Exercise 2.1.25. 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 following probabilities: a. P(A ∪ B)
> Transactions to a computer database are either new items or changes to previous items. The addition of an item can be completed in less than 100 milliseconds 90% of the time, but only 20% of changes to a previous item can be completed in less than this t
> A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Assume all passwords are equally likely. Let A and B denote the events
> A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Let A denote the event that the design color i
> A computer system uses passwords that are six characters, and each character is one of the 26 letters (a–z) or 10 integers (0–9). Uppercase letters are not used. Let A denote the event that a password begins with a vowel (either a, e, i, o, or u), and le
> Strands of copper wire from a manufacturer are analyzed for strength and conductivity. The results from 100 strands are as follows: a. If a strand is randomly selected, what is the probability that its conductivity is high and its strength is high? b. If
> Consider the hospital emergency room data in Example 2.6. 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). Use the addition rules to calculate the following probabilities. a
> In the article “ACL Reconstruction Using Bone-Patellar Tendon-Bone Press-Fit Fixation: 10-Year Clinical Results” in Knee Surgery, Sports Traumatology, Arthroscopy (2005, Vol. 13, pp. 248–255), the fol
> 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: a. Find the probability t
> If P(A) = 0.3, P(B) = 0.2, and P(A ∩ B) = 0.1, determine the following probabilities: a. P(A′) b. P(A ∪ B) c. P(A′ ∩ B) d. P(A ∩ B′) e. P[(A ∪ B)′] f. P(A′ ∪ B)
> If A, B, and C are mutually exclusive events with P(A) = 0.2, P(B) = 0.3, and P(C) = 0.4, determine the following probabilities: a. P(A ∪ B ∪ C) b. P(A ∩ B ∩ C) c. P(A ∩ B) d. P[(A ∪ B) ∩ C] e. P(A′ ∩ B′ ∩ C′)
> 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 sam
> In circuit testing of printed circuit boards, each board either fails or does not fail the test. A board that fails the test is then checked further to determine which one of five defect types is the primary failure mode. Represent the sample space for t
> 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 (Transac
> 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
> 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
> 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 sel
> 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;
> 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
> 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?
> 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 cavi
> 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?
> 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 ∪
> The probability that a customer’s order is not shipped on time is 0.05. A particular customer places three orders, and the orders are placed far enough apart in time that they can be considered to be independent events. a. What is the probability that al
> Similar to the hospital schedule in Example 2.9, suppose 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 shoul
> 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 selects 3 parts wi
> A bin of 50 parts contains 5 that are defective. A sample of 10 parts is selected without replacement. How many samples contain at least 4 defective parts?
> 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
> 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
> 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 pl
> 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 sequen
> 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?
> Abatch of 140 semiconductor chips is inspected by choosing a sample of 5 chips without replacement. Assume 10 of the chips do not conform to customer requirements. a. How many different samples are possible? b. How many samples contain exactly one noncon
> A manufacturing process consists of 10 operations that can be completed in any order. How many different production sequences are possible?
> Energy released from cells breaks the molecular bond and converts ATP (adenosine triphosphate) into ADP (adenosine diphosphate). Storage of ATP in muscle cells (even for an athlete) can sustain maximal muscle power only for less than five seconds (a shor
> 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?
> 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 fol
> 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?
> 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
> 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
> 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 ith bit is distorted, i = 1,…, 4. a. Describe the sample space for this experiment. b. Are the
> 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 blu
> 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 d
> 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 inc
> Three events are shown on the Venn diagram in the following figure: Reproduce the figure and shade the region that corresponds to each of the following events. a. A′ b. A ∩ B c. (A ∩ B) â
> An electronic storage device uses an error recovery procedure that requires an immediate satisfactory readback of any written data. If the readback is not successful after three writing operations, that sector of the device is eliminated as unacceptable
> Three attempts are made to read data in a magnetic storage device before an error recovery procedure is used. The error recovery procedure attempts three corrections before an “abort” message is sent to the operator. Let s denote the success of a read op
> Calls are repeatedly placed to a busy phone line until a connection is achieved.
> A sampled injection-molded part could have been produced in either one of two presses and in any one of the eight cavities in each press.
> 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.
> The pH reading of a water sample to the nearest tenth of a unit.
> The time until a service transaction is requested of a computer to the nearest millisecond.
> Consider the treatments in Exercise 2.1.25. Suppose a patient is selected randomly. Let A denote the event that the patient is treated with ribavirin plus interferon alfa or interferon alfa, and let B denote the event that the response is complete. Deter
> A congested computer network has a 0.002 probability of losing a data packet, and packet losses are independent events. A lost packet must be resent. a. What is the probability that an e-mail message with 100 packets will need to be resent? b. What is th
> A batch contains 36 bacteria cells. Assume that 12 of the cells are not capable of cellular replication. Of the cells, 6 are selected at random, without replacement, to be checked for replication. a. What is the probability that all 6 of the selected cel
> An article in Genome Research [“An Assessment of Gene Prediction Accuracy in Large DNA Sequences” (2000, Vol. 10, pp. 1631–1642)] considered the accuracy of commercial software to predict nucleotides
> A company that tracks the use of its Web site determined that the more pages a visitor views, the more likely the visitor is to request more information (RMI). Use the following table to answer the questions: a. What is the probability that a visitor to
> A machine tool is idle 15% of the time. You request immediate use of the tool on five different occasions during the year. Assume that your requests represent independent events. a. What is the probability that the tool is idle at the time of all of your
> A robotic insertion tool contains 10 primary components. The probability that any component fails during the warranty period is 0.01. Assume that the components fail independently and that the tool fails if any component fails. What is the probability th
> In the manufacturing of a chemical adhesive, 3% of all batches have raw materials from two different lots. This occurs when holding tanks are replenished and the remaining portion of a lot is insufficient to fill the tanks. Only 5% of batches with materi
> Incoming calls to a customer service center are classified as complaints (75% of calls) or requests for information (25% of calls). Of the complaints, 40% deal with computer equipment that does not respond and 57% deal with incomplete software installati
> 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?
> A lot contains 15 castings from a local supplier and 25 castings from a supplier in the next state. Two castings are selected randomly, without replacement, from the lot of 40. Let A be the event that the first casting selected is from the local supplier
> Samples of a cast aluminum part are classified on the basis of surface finish (in microinches) and edge finish. The results of 100 parts are summarized as follows: Let A denote the event that a sample has excellent surface finish, and let B denote the e
> Samples of laboratory glass are in small, light packaging or heavy, large packaging. Suppose that 2% and 1%, respectively, of the sample shipped in small and large packages, respectively, break during transit. If 60% of the samples are shipped in large p
> Decide whether a discrete or continuous random variable is the best model for each of the following variables: a. The time for a computer algorithm to assign an image to a category. b. The number of bytes used to store a file in a computer. c. The ozone
> Decide whether a discrete or continuous random variable is the best model for each of the following variables: a. The time until a projectile returns to earth. b. The number of times a transistor in a computer memory changes state in one operation. c. Th
> Decide whether a discrete or continuous random variable is the best model for each of the following variables: a. The number of cracks exceeding one-half inch in 10miles of an interstate highway. b. The weight of an injection-molded plastic part. c. The
> The following circuit operates if and only if there is a path of functional devices from left to right. Assume that devices fail independently and that the probability of failure of each device is as shown. What is the probability that the circuit does n
> A computer system uses passwords that contain exactly eight characters, and each character is one of 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let Ω denote the set of all possible passwords, and let A and B denote the
> Table 2.1 provides data on wafers categorized by location and contamination levels. Let A denote the event that contamination is low, and let B denote the event that the location is center. Are A and B independent? Why or why not? Data from Table 2.1:
> An integrated circuit contains 10 million logic gates (each can be a logical AND or OR circuit). Assume the probability of a gate failure is p and that the failures are independent. The integrated circuit fails to function if any gate fails. Determine th
> A Web ad can be designed fromfour different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Let A denote the event that the design color is
> Consider the hospital emergency room data in Example 2.6. 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). Are these events independent? Data from Example 2.6:
> The following circuit operates if and only if there is a path of functional devices from left to right. The probability that each device functions is as shown. Assume that the probability that a device is functional does not depend on whether or not othe
> The following circuit operates if and only if there is a path of functional devices from left to right. The probability that each device functions is as shown. Assume that the probability that a device is functional does not depend on whether or not othe
> A credit card contains 16 digits. It also contains the month and year of expiration. Suppose there are 1 million credit card holders with unique card numbers. A hacker randomly selects a 16-digit credit card number. a. What is the probability that it bel
> An e-mail filter is planned to separate valid e-mails from spam. The word free occurs in 60% of the spam messages and only 4% of the valid messages. Also, 20% of the messages are spam. Determine the following probabilities: a. The message contains free.
> A recreational equipment supplier finds that among orders that include tents, 40% also include sleeping mats. Only 5% of orders that do not include tents do include sleeping mats. Also, 20% of orders include tents. Determine the following probabilities:
> An encryption–decryption system consists of three elements: encode, transmit, and decode. A faulty encode occurs in 0.5% of the messages processed, transmission errors occur in 1% of the messages, and a decode error occurs in 0.1% of the messages. Assume
> Two Web colors are used for a site advertisement. If a site visitor arrives from an affiliate, the probabilities of the blue or green colors being used in the advertisement are 0.8 and 0.2, respectively. If the site visitor arrives from a search site, th
> Suppose that a patient is selected randomly from those described in Exercise 2.4.11. Let A denote the event that the patient is in group 1, and let B denote the event that there is no progression. Determine the following probabilities: a. P(B) b. P(B |
> Consider the hospital emergency room data in Example 2.6. Use Bayes’ theorem to calculate the probability that a person visits hospital 4 given they are LWBS. Data from Example 2.6:
> Software to detect fraud in consumer phone cards tracks the number of metropolitan areas where calls originate each day. It is found that 1% of the legitimate users originate calls from two or more metropolitan areas in a single day. However, 30% of frau
> 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 impo
> An ammeter that displays three digits is used to measure current in milliamperes.
> 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.
> The number of hits (views) is recorded at a high-volume Web site in a day.
> Each of four transmitted bits is classified as either in error or not in error.
> 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 Ω denote the set of all possible passwords, and let A and B denote t
> It is known that two defective cellular phones were erroneously sent to a shipping lot that now has a total of 75 phones. A sample of phones will be selected from the lot without replacement. a. If three phones are inspected, determine the probability th
> The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of Chronic Hepatitis C” [Gastroenterology (1996, Vol. 111(5), pp. 1307–1312)], considered the effect of two treatments and
> Consider the hospital emergency department data in Example 2.6. 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
> 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?
> 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 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
> 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 sam
> In control replication, cells are replicated over a period of two days. 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
> Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized here: Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high sc
