Samples of emissions from three suppliers are classified for conformance to air-quality

Each of three machined parts is classified as either above or below the target specification for the part.

Each of four transmitted bits is classified as either in error or not in error.

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 24 Web sites is classified as containing or not containing banner ads.

An ammeter that displays three digits is used to measure current in milliamperes.

A scale that displays two decimal places is used to measure material feeds in a chemical plant in tons.

The following two questions appear on an employee survey questionnaire. Each answer is chosen from the fivepoint 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?

The concentration of ozone to the nearest part per billion.

The time until a service transaction is requested of a computer to the nearest millisecond.

The pH reading of a water sample to the nearest tenth of a unit.

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 time of a chemical reaction is recorded to the nearest millisecond.

An order for an automobile can specify either an automatic or a standard transmission, either with or without air-conditioning, and any one of the four colors red, blue, black, or white. Describe the set of possible orders for this experiment.

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.

An order for a computer system can specify memory of 4, 8, or 12 gigabytes, and disk storage of 200, 300, or 400 gigabytes. Describe the set of possible orders.

Calls are repeatedly placed to a busy phone line until a connection is achieved.

In a magnetic storage device, three attempts are made to read data 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 F denote the failure of an error recovery procedure S denote the success 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.

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.

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 digital scale is used that provides weights to the nearest gram. (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.

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) (c) (d) (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? 2-23.

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, . (a) Describe the sample space for this experiment. (b) Are the Ai s mutually exclusive? Describe the outcomes in each of the following events:

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 675700 nm and the blue range is 450500 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:

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 identifiedone 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) (d) A B A B

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized below: shock resistance high low scratch high 70 9 resistance low 16 5 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 and.

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: edge finish excellent good surface excellent 80 2 finish good 10 8 (a) Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent edge finish. Determine the number of samples in and . (b) Assume that each of two samples is to be classified on the basis of surface finish, either excellent or good, and on the basis of edge finish, either excellent or good. Use a tree diagram to represent the possible outcomes of this experiment.

conformance to air-quality specifications. The results from 100 samples are summarized as follows: conforms yes no 1 22 8 supplier 2 25 5 3 30 10 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 and

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: and . Describe each of the following events: (a) (b) (c) (d) A B A B

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 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, eight boards with minor defects, and two boards with major defects. (b) The batch contains 90 boards that are not defective, eight boards with minor defects, and one board with major defects.

Counts of the Web pages provided by each of two computer servers in a selected hour of the day are recorded. Let A denote the event that at least 10 pages are provided by server 1 and let B denote the event that at least 20 pages are provided by server 2. (a) Describe the sample space for the numbers of pages for the two servers graphically in an plot. Show each of the following events on the sample space graph: (b) A (c) B (d) (e)

The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space for the rise time of each batch be positive, real numbers. Consider the rise times of two batches. Let A denote the event that the rise time of batch 1 is less than 72.5 minutes, and let B denote the event that the rise time of batch 2 is greater than 52.5 minutes. Describe the sample space for the rise time of two batches graphically and show each of the following events on a twodimensional plot: (a) A (b) (c) (d)

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?

An order for a computer can specify any one of five memory sizes, any one of three types of displays, and any one of four sizes of a hard disk, and can either include or not include a pen tablet. How many different systems can be ordered?

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?

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?

A manufacturing process consists of 10 operations that can be completed in any ord

A manufacturing operation consists of 10 operations. However, five machining operations must be completed before any of the remaining five assembly operations can begin. Within each set of five, operations can be completed in any order. How many different production sequences are possible?

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?

A batch of 140 semiconductor chips is inspected by choosing a sample of five 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?

In the layout of a printed circuit board for an electronic product, there are 12 different locations that 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?

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 two times 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. (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?

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?

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 byte is a sequence of eight bits and each bit is either 0 or 1. (a) How many different bytes are possible? (b) If the first bit of a byte is a parity check, that is, the first byte is determined from the other seven bits, how many different bytes are possible?

In a chemical plant, 24 holding tanks are used for final product storage. Four tanks are selected at random and without replacement. Suppose that six of the tanks contain material in which the viscosity exceeds the customer requirements. (a) What is the probability that exactly one tank in the sample contains high-viscosity material? (b) What is the probability that at least one tank in the sample contains high-viscosity material? (c) In addition to the six tanks with high viscosity levels, four different tanks contain material with high impurities. What is the probability that exactly one tank in the sample contains high-viscosity material and exactly one tank in the sample contains material with high impurities?

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 three 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) What is the probability that the inspector finds exactly one nonconforming part? (b) What is the probability that the inspector finds at least one nonconforming part?

A bin of 50 parts contains five that are defective. A sample of two parts is selected at random, without replacement. Determine the probability that both parts in the sample are defective.

The following table summarizes 204 endothermic reactions involving sodium bicarbonate. Final Temperature Heat Absorbed (cal) Conditions Below Target Above Target 266 K 12 40 271 K 44 16 274 K 56 36 Wells Geological Formation Group Failed Total Gneiss 170 1685 Granite 2 28 Loch raven schist 443 3733 Mafic 14 363 Marble 29 309 Prettyboy schist 60 1403 Other schists 46 933 Serpentine 3 39 Let A denote the event that a reaction 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) (b) A (c) (d) (e)

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?

Consider the hospital emergency department data in Example 2-8. 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) (b) A (c) (d) (e)

An article in The Journal of Data Science [A Statistical Analysis of Well Failures in Baltimore County (2009, Vol. 7, pp. 111127)] provided the following table of well failures for different geological formation groups in Baltimore County. A B A B A B A B A B A B A B A B Let A denote the event that the geological formation has more than 1000 wells and let B denote the event that a well failed. Determine the number of wells in each of the following events. (a) (b) A B A (c) (d) (e) A B A B A B

Each of the possible five outcomes of a random experiment is equally likely. The sample space is {a, b, c, d, e}. Let A denote the event {a, b}, and let B denote the event {c, d, e}. Determine the following: (a) (b) (c) (d) (e)

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) (b) (c) (d) (e)

Orders for a computer are summarized by the optional features that are requested as follows: proportion of orders no optional features 0.3 one optional feature 0.5 more than one optional feature 0.2 (a) What is the probability that an order requests at least one optional feature? (b) What is the probability that an order does not request more than one optional feature?

If the last digit of a weight measurement is equally likely to be any of the digits 0 through 9, (a) What is the probability that the last digit is 0? (b) What is the probability that the last digit is greater than or equal to 5?

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?

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 a part is from cavity 1 or 2? (c) What is the probability that a part is from neither cavity 3 nor 4?

In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Because acids and bases are usually colorless (as are the water and salt produced in the neutralization reaction), pH is measured to monitor the reaction. Suppose that the equivalence point is reached after approximately 100 mL of a NaOH solution have been added (enough to react with all the acetic acid present) but that replicates are equally likely to indicate from 95 to 104 mL to the nearest mL. Assume that volumes are measured to the nearest mL and describe the sample space. (a) What is the probability that equivalence is indicated at 100 mL? (b) What is the probability that equivalence is indicated at less than 100 mL? (c) What is the probability that equivalence is indicated between 98 and 102 mL (inclusive)?

In a NiCd battery, a fully charged cell is composed of Nickelic Hydroxide. Nickel is an element that has multiple oxidation states that is usually found in the following states: nickel charge proportions found 0 0.17 2 0.35 3 0.33 4 0.15 (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?

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?

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?

A message can follow different paths through servers on a network. The senders 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 recipients 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?

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 the previous question change if five or six or seven different amounts are used in the first step? Explain.

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: shock resistance high low scratch high 70 9 resistance low 16 5 . 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) (b) (c) (d) (e) (f)

Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows: conforms yes no 1 22 8 supplier 2 25 5 3 30 10 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) (b) (c) (d) (e) (f)

Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows: conforms yes no 1 22 8 supplier 2 25 5 3 30 10 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) (b) (c) (d) (e) (f)

Use the axioms of probability to show the following: (a) For any event . (b) (c) If A is contained in B, then

Consider the endothermic reactions in Exercise 2-50. Let A denote the event that a reaction final temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Determine the following probabilities. (a) P (b) P(A) (c) P (d) P1A B2 (e) P1A B2 1A B2 1A B2

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. If you visit the site five times, what is the probability that you will not see the same design?

Consider the hospital emergency room data in Example 2-8. 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). Determine the following probabilities.

Consider the well failure data in Exercise 2-53. Let A denote the event that the geological formation has more than 1000 wells and let B denote the event that a well failed. Determine the following probabilities. (a) P (b) P(A) (c) P (d) P1A B2 (e) P1A B2

If , and determine the following probabilities: (a) (b) (c) (d) (e) (f)

If A, B, and C are mutually exclusive events with and determine the following probabilities: (a) (b) (c) (d) (e)

In the article ACL Reconstruction Using BonePatellar Tendon-Bone Press-Fit Fixation: 10-Year Clinical Results in Knee Surgery, Sports Traumatology, Arthroscopy (2005, Vol. 13, pp. 248255), the following causes for knee injuries were considered: Percentage of Activity Knee Injuries Contact sport 46% Noncontact sport 44% Activity of daily living 9% Riding motorcycle 1% (a) What is the probability a knee injury resulted from a sport (contact or noncontact)? (b) What is the probability a knee injury resulted from an activity other than a sport?

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: shock resistance high low scratch high 70 9 resistance low 16 5 (a) If a disk is selected at random, what is the probability that its scratch resistance is high and its shock resistance is high? (b) If a disk is selected at random, what is the probability that its scratch resistance is high or its shock resistance is high?(c) Consider the event that a disk has high scratch resistance and the event that a disk has high shock resistance. Are these two events mutually exclusive?

Strands of copper wire from a manufacturer are analyzed for strength and conductivity. The results from 100 strands are as follows: strength high low high conductivity 74 8 low conductivity 15 3 (a) If a strand is randomly selected, what is the probability that its conductivity is high and its strength is high? (b) If a strand is randomly selected, what is the probability that its conductivity is low or the strength is low? (c) Consider the event that a strand has low conductivity and the event that the strand has a low strength. Are these two events mutually exclusive?

The analysis of shafts for a compressor is summarized by conformance to specifications. roundness conforms yes no surface finish yes 345 5 conforms no 12 8 (a) If a shaft is selected at random, what is the probability that the shaft conforms to surface finish requirements? (b) What is the probability that the selected shaft conforms to surface finish requirements or to roundness requirements? (c) What is the probability that the selected shaft either conforms to surface finish requirements or does not conform to roundness requirements? (d) What is the probability that the selected shaft conforms to both surface finish and roundness requirements?

Cooking oil is produced in two main varieties: monoand polyunsaturated. Two common sources of cooking oil are corn and canola. The following table shows the number of bottles of these oils at a supermarket: type of oil canola corn type of mono 7 13 unsaturation poly 93 77 (a) If a bottle of oil is selected at random, what is the probability that it belongs to the polyunsaturated category? (b) What is the probability that the chosen bottle is monounsaturated canola oil?

A manufacturer of front lights for automobiles tests lamps under a high-humidity, high-temperature environment using intensity and useful life as the responses of interest. The following table shows the performance of 130 lamps: useful life satisfactory unsatisfactory intensity satisfactory 117 3 unsatisfactory 8 2 (a) Find the probability that a randomly selected lamp will yield unsatisfactory results under any criteria. (b) The customers for these lamps demand 95% satisfactory results. Can the lamp manufacturer meet this demand?

A computer system uses passwords that are six characters and each character is one of the 26 letters (az) or 10 integers (09). 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 let B denote the event that a password ends with an even number (either 0, 2, 4, 6, or 8). Suppose a hacker selects a password at random. Determine the following probabilities:(a) (b) (c) (d)

Consider the endothermic reactions in Exercise 2-50. Let A denote the event that a reaction final temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Use the addition rules to calculate the following probabilities. (a) (b) (c)

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 is red and let B denote the event that the font size is not the smallest one. Use the addition rules to calculate the following probabilities. (a) (b) (c)

Consider the hospital emergency room data in Example 2-8. 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) (b) (c)

Consider the well failure data in Exercise 2-53. Let A denote the event that the geological formation has more than 1000 wells and let B denote the event that a well failed. Use the addition rules to calculate the following probabilities. (a) (b) (c) P1A B2 P1A B2 P1A B2

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: shock resistance high low scratch high 70 9 resistance low 16 5 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 following probabilities: (a) P(A) (b) P(B) (c) (d)

Samples of skin experiencing desquamation are analyzed for both moisture and melanin content. The results from 100 skin samples are as follows: melanin content high low moisture high 13 7 content low 48 32 Let A denote the event that a sample has low melanin content, and let B denote the event that a sample has high moisture content. Determine the following probabilities: (a) P(A) (b) P(B) (c) P(A B) (d)

The analysis of results from a leaf transmutation experiment (turning a leaf into a petal) is summarized by type of transformation completed: total textural transformation yes no total color yes 243 26 transformation no 13 18 (a) If a leaf completes the color transformation, what is the probability that it will complete the textural transformation? (b) If a leaf does not complete the textural transformation, what is the probability it will complete the color transformation

Samples of a cast aluminum part are classified on the basis of surface finish (in microinches) and length measurements. The results of 100 parts are summarized as follows: length excellent good surface excellent 80 2 finish good 10 8 P1A B2 P1B A2 Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent length. Determine: (a) (b) (c) (d) (e) If the selected part has excellent surface finish, what is the probability that the length is excellent? (f ) If the selected part has good length, what is the probability that the surface finish is excellent?

The following table summarizes the analysis of samples of galvanized steel for coating weight and surface roughness: coating weight high low surface high 12 16 roughness low 88 34 (a) If the coating weight of a sample is high, what is the probability that the surface roughness is high? (b) If the surface roughness of a sample is high, what is the probability that the coating weight is high? (c) If the surface roughness of a sample is low, what is the probability that the coating weight is low?

Consider the data on wafer contamination and location in the sputtering tool shown in Table 2-2. Assume that one wafer is selected at random from this set. Let A denote the event that a wafer contains four or more particles, and let B denote the event that a wafer is from the center of the sputtering tool. Determine: (a) (b) (c) (d) (e) (f)

The following table summarizes the number of deceased beetles under autolysis (the destruction of a cell after its death by the action of its own enzymes) and putrefaction (decomposition of organic matter, especially protein, by microorganisms, resulting in production of foul-smelling matter): autolysis high low putrefaction high 14 59 low 18 9 (a) If the autolysis of a sample is high, what is the probability that the putrefaction is low? (b) If the putrefaction of a sample is high, what is the probability that the autolysis is high? (c) If the putrefaction of a sample is low, what is the probability that the autolysis is low?

A maintenance firm has gathered the following information regarding the failure mechanisms for air conditioning systems: evidence of gas leaks yes no evidence of yes 55 17 electrical failure no 32 3 The units without evidence of gas leaks or electrical failure showed other types of failure. If this is a representative sample of AC failure, find the probability (a) That failure involves a gas leak (b) That there is evidence of electrical failure given that there was a gas leak (c) That there is evidence of a gas leak given that there is evidence of electrical failure

A lot of 100 semiconductor chips contains 20 that are defective. Two are selected randomly, without replacement, from the lot. (a) What is the probability that the first one selected is defective? (b) What is the probability that the second one selected is defective given that the first one was defective? (c) What is the probability that both are defective? (d) How does the answer to part (b) change if chips selected were replaced prior to the next selection?

A batch of 500 containers for frozen orange juice contains five that are defective. Two are selected, at random, without replacement from the batch. (a) What is the probability that the second one selected is defective given that the first one was defective? (b) What is the probability that both are defective? (c) What is the probability that both are acceptable? Three containers are selected, at random, without replacement, from the batch. (d) What is the probability that the third one selected is defective given that the first and second ones selected were defective? (e) What is the probability that the third one selected is defective given that the first one selected was defective and the second one selected was okay? (f ) What is the probability that all three are defective?

A batch of 350 samples of rejuvenated mitochondria contains eight that are mutated (or defective). Two are selected, at random, without replacement from the batch. (a) What is the probability that the second one selected is defective given that the first one was defective? (b) What is the probability that both are defective? (c) What is the probability that both are acceptable?

A computer system uses passwords that are exactly seven characters and each character is one of the 26 letters (az) or 10 integers (09). You maintain a password for this computer system. Let A denote the subset of passwords that begin with a vowel (either a, e, i, o, or u) and let B denote the subset of passwords that end with an even number (either 0, 2, 4, 6, or 8). (a) Suppose a hacker selects a password at random. What is the probability that your password is selected? (b) Suppose a hacker knows your password is in event A and selects a password at random from this subset. What is the probability that your password is selected? (c) Suppose a hacker knows your password is in A and B and selects a password at random from this subset. What is the probability that your password is selected?

If , must A B? Draw a Venn diagram to explain your answer.

Suppose A and B are mutually exclusive events. Construct a Venn diagram that contains the three events A, B, and C such that and .

Consider the endothermic reactions in Exercise 2-50. Let A denote the event that a reaction final temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Determine the following probabilities. (a) (b) (c) (d)

Consider the hospital emergency room data in Example 2-8. 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). Determine the following probabilities. (a) (b) (c) (d)

Consider the well failure data in Exercise 2-53. (a) What is the probability of a failure given there are more than 1000 wells in a geological formation? (b) What is the probability of a failure given there are fewer than 500 wells in a geological formation?

An article in the The Canadian Entomologist (Harcourt et al., 1977, Vol. 109, pp. 15211534) studied the life of the alfalfa weevil from eggs to adulthood. The following table shows the number of larvae that survived at each stage of development from eggs to adults. Early Late Pre- Late Eggs Larvae Larvae pupae Pupae Adults 421 412 306 45 35 31 (a) What is the probability an egg survives to an adult? (b) What is the probability of survival to adult given survival to the late larvae stage? (c) What stage has the lowest probability of survival to the next stage?

Suppose that and Determine the following: (a) (b)

Suppose that and What is P(A)?

The probability is 1% that an electrical connector that is kept dry fails during the warranty period of a portable computer. If the connector is ever wet, the probability of a failure during the warranty period is 5%. If 90% of the connectors are kept dry and 10% are wet, what proportion of connectors fail during the warranty period?

Suppose 2% of cotton fabric rolls and 3% of nylon fabric rolls contain flaws. Of the rolls used by a manufacturer, 70% are cotton and 30% are nylon. What is the probability that a randomly selected roll used by the manufacturer contains flaws?

The edge roughness of slit paper products increases as knife blades wear. Only 1% of products slit with new blades have rough edges, 3% of products slit with blades of average sharpness exhibit roughness, and 5% of products slit with worn blades exhibit roughness. If 25% of the blades in manufacturing are new, 60% are of average sharpness, and 15% are worn, what is the proportion of products that exhibit edge roughness?

In the 2004 presidential election, exit polls from the critical state of Ohio provided the following results: total Bush, 2004 Kerry, 2004 no college degree (62%) 50% 50% college graduate (38%) 53% 46% What is the probability a randomly selected respondent voted for Bush?

Computer keyboard failures are due to faulty electrical connects (12%) or mechanical defects (88%). Mechanical defects are related to loose keys (27%) or improper assembly (73%). Electrical connect defects are caused by defective P1B2 0.8. P1A B2 0.2, P1A B2 0.3, P1A B2 P1A B2 P1A B2 0.4 P1B2 0.5. wires (35%), improper connections (13%), or poorly welded wires (52%). (a) Find the probability that a failure is due to loose keys. (b) Find the probability that a failure is due to improperly connected or poorly welded wires. 2-1

Heart failures are due to either natural occurrences (87%) or outside factors (13%). Outside factors are related to induced substances (73%) or foreign objects (27%). Natural occurrences are caused by arterial blockage (56%), disease (27%), and infection (e.g., staph infection) (17%). (a) Determine the probability that a failure is due to induced substance. (b) Determine the probability that a failure is due to disease or infection.

A batch of 25 injection-molded parts contains five that have suffered excessive shrinkage. (a) If two parts are selected at random, and without replacement, what is the probability that the second part selected is one with excessive shrinkage? (b) If three parts are selected at random, and without replacement, what is the probability that the third part selected is one with excessive shrinkage?

A lot of 100 semiconductor chips contains 20 that are defective. (a) Two are selected, at random, without replacement, from the lot. Determine the probability that the second chip selected is defective. (b) Three are selected, at random, without replacement, from the lot. Determine the probability that all are defective.

An article in the British Medical Journal [Comparison of Treatment of Renal Calculi by Operative Surgery, Percutaneous Nephrolithotomy, and Extracorporeal Shock Wave Lithotripsy (1986, Vol. 82, pp. 879892)] provided the following discussion of success rates in kidney stone removals. Open surgery had a success rate of 78% (273/350) while a newer method, percutaneous nephrolithotomy (PN),had a success rate of 83% (289/350). This newer method looked better, but the results changed when stone diameter was considered. For stones with diameters less than two centimeters, 93% (81/87) of cases of open surgery were successful compared with only 83% (234/270) of cases of PN. For stones greater than or equal to two centimeters, the success rates were 73% (192/263) and 69% (55/80) for open surgery and PN, respectively. Open surgery is better for both stone sizes, but less successful in total. In 1951, E. H. Simpson pointed out this apparent contradiction (known as Simpsons Paradox) but the hazard still persists today. Explain how open surgery can be better for both stone sizes but worse in total.

Consider the endothermic reactions in Exercise 2-50. Let A denote the event that a reaction final temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Determine the following probabilities. (a) (b) (c) (d) Use the total probability rule to determine P(A)

Consider the hospital emergency room data in Example 2-8. 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). Determine the following probabilities. (a) (b) (c) (d) Use the total probability rule to determine P(A)

Consider the hospital emergency room data in Example 2-8. Suppose that three visits that resulted in LWBS are selected randomly (without replacement) for a follow-up interview. (a) What is the probability that all three are selected from Hospital 2? (b) What is the probability that all three are from the same hospital?

Consider the well failure data in Exercise 2-53. Let A denote the event that the geological formation has more than 1000 wells and let B denote the event that a well failed. Determine the following probabilities. (a) (b) (c) (d) Use the total probability rule to determine P(A)

Consider the well failure data in Exercise 2-53. Suppose that two failed wells are selected randomly (without replacement) for a follow-up review. (a) What is the probability that both are from the gneiss geological formation group? (b) What is the probability that both are from the same geological formation group?

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 in randomly generated by the Web server when you visit the site. Determine the probability that the ad color is red and the font size is not the smallest one.

If and are the events A and B independent?

If and are the events B and the complement of A independent?

If and A and B are mutually exclusive, are they independent?

A batch of 500 containers for frozen orange juice contains five that are defective. Two are selected, at random, without replacement, from the batch. Let A and B denote the events that the first and second containers selected are defective, respectively. (a) Are A and B independent events? (b) If the sampling were done with replacement, would A and B be independent?

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: shock resistance high low scratch high 70 9 resistance low 16 5 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. Are events A and B independent?

Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows: conforms yes no 1 22 8 supplier 2 25 5 3 30 10 Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications. (a) Are events A and B independent? (b) Determine

Redundant Array of Inexpensive Disks (RAID) is a technology that uses multiple hard drives to increase the speed of data transfer and provide instant data backup. Suppose that the probability of any hard drive failing in a day is 0.001 and the drive failures are independent. (a) A RAID 0 scheme uses two hard drives, each containing a mirror image of the other. What is the probability of data loss? Assume that data loss occurs if both drives fail within the same day. (b) A RAID 1 scheme splits the data over two hard drives. What is the probability of data loss? Assume that data loss occurs if at least one drive fails within the same day.

The probability that a lab specimen contains high levels of contamination is 0.10. Five samples are checked, and the samples are independent. (a) What is the probability that none contains high levels of contamination? (b) What is the probability that exactly one contains high levels of contamination? (c) What is the probability that at least one contains high levels of contamination?

In a test of a printed circuit board using a random test pattern, an array of 10 bits is equally likely to be 0 or 1. Assume the bits are independent. (a) What is the probability that all bits are 1s? (b) What is the probability that all bits are 0s? (c) What is the probability that exactly five bits are 1s and five bits are 0s?

Six tissues are extracted from an ivy plant infested by spider mites. The plant in infested in 20% of its area. Each tissue is chosen from a randomly selected area on the ivy plant. (a) What is the probability that four successive samples show the signs of infestation? (b) What is the probability that three out of four successive samples show the signs of infestation?

A player of a video game is confronted with a series of four opponents and an 80% probability of defeating each opponent. Assume that the results from opponents are independent (and that when the player is defeated by an opponent the game ends). (a) What is the probability that a player defeats all four opponents in a game? (b) What is the probability that a player defeats at least two opponents in a game? (c) If the game is played three times, what is the probability that the player defeats all four opponents at least once?

In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Since acids and bases are usually colorless (as are the water and salt produced in the neutralization reaction), pH is measured to monitor the reaction. Suppose that the equivalence point is reached after approximately 100 mL of a NaOH solution has been added (enough to react with all the acetic acid present) but that replicates are equally likely to indicate from 95 to 104 mL, measured to the nearest mL. Assume that two technicians each conduct titrations independently. (a) What is the probability that both technicians obtain equivalence at 100 mL? (b) What is the probability that both technicians obtain equivalence between 98 and 104 mL (inclusive)? (c) What is the probability that the average volume at equivalence from the technicians is 100 mL?

A credit card contains 16 digits. It also contains a month and year of expiration. Suppose there are one 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 belongs to a user? (b) Suppose a hacker has a 25% chance of correctly guessing the year your card expires and randomly selects one of the 12 months. What is the probability that the hacker correctly selects the month and year of expiration?

Eight cavities in an injection-molding tool produce plastic connectors that fall into a common stream. A sample is chosen every several minutes. Assume that the samples are independent. (a) What is the probability that five successive samples were all produced in cavity one of the mold? (b) What is the probability that five successive samples were all produced in the same cavity of the mold? (c) What is the probability that four out of five successive samples were produced in cavity one of the mold?

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 other devices are functional. What is the probability that the circuit operates?

The following circuit operates if and only if there is a path of functional devices from left to right. The probability each device functions is as shown. Assume that the probability that a device functions does not depend on whether or not other devices are functional. What is the probability that the circuit operates?

Consider the endothermic reactions in Exercise 2-50. Let A denote the event that a reaction final temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Are these events independent?

Consider the hospital emergency room data in Example 2-8. 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?

Consider the well failure data in Exercise 2-53. Let A denote the event that the geological formation has more than 1000 wells and let B denote the event that a well failed. Are these events independent?

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 is red and Let B denote the event that the font size is not the smallest one. Are A and B independent events? Explain why or why not.

Suppose that and Determine

Suppose that and P1B2 0.8. Determine P1

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 fraudulent users originate calls from two or more metropolitan areas in a single day. The proportion of fraudulent users is 0.01%. If the same user originates calls from two or more metropolitan areas in a single day, what is the probability that the user is fraudulent?

A new process of more accurately detecting anaerobic respiration in cells is being tested. The new process is important due to its high accuracy, its lack of extensive experimentation, and the fact that it could be used to identify five different categories of organisms: obligate anaerobes, facultative anaerobes, aerotolerant, microaerophiles, and nanaerobes instead of using a single test for each category. The process claims that it can identify obligate anaerobes with 97.8% accuracy, facultative anaerobes with 98.1% accuracy, aerotolerant with 95.9% accuracy, microaerophiles with 96.5% accuracy, and nanaerobes with 99.2% accuracy. If any category is not present, the process does not signal. Samples are prepared for the calibration of the process and 31% of them contain obligate anaerobes, 27% contain facultative anaerobes, 21% contain microaerophiles, 13% contain nanaerobes, and 8% contain aerotolerant. A test sample is selected randomly. (a) What is the probability that the process will signal? (b) If the test signals, what is the probability that microaerophiles are present?

In the 2004 presidential election, exit polls from the critical state of Ohio provided the following results: Bush Kerry no college degree (62%) 50% 50% college graduate (38%) 53% 46% If a randomly selected respondent voted for Bush, what is the probability that the person has a college degree?

Customers are used to evaluate preliminary product designs. In the past, 95% of highly successful products received good reviews, 60% of moderately successful products received good reviews, and 10% of poor products received good reviews. In addition, 40% of products have been highly successful, 35% have been moderately successful, and 25% have been poor products. (a) What is the probability that a product attains a good review? (b) If a new design attains a good review, what is the probability that it will be a highly successful product?(c) If a product does not attain a good review, what is the probability that it will be a highly successful product?

An inspector working for a manufacturing company has a 99% chance of correctly identifying defective items and a 0.5% chance of incorrectly classifying a good item as defective. The company has evidence that its line produces 0.9% of nonconforming items. (a) What is the probability that an item selected for inspection is classified as defective? (b) If an item selected at random is classified as nondefective, what is the probability that it is indeed good?

A new analytical method to detect pollutants in water is being tested. This new method of chemical analysis is important because, if adopted, it could be used to detect three different contaminantsorganic pollutants, volatile solvents, and chlorinated compoundsinstead of having to use a single test for each pollutant. The makers of the test claim that it can detect high levels of organic pollutants with 99.7% accuracy, volatile solvents with 99.95% accuracy, and chlorinated compounds with 89.7% accuracy. If a pollutant is not present, the test does not signal. Samples are prepared for the calibration of the test and 60% of them are contaminated with organic pollutants, 27% with volatile solvents, and 13% with traces of chlorinated compounds. A test sample is selected randomly. (a) What is the probability that the test will signal? (b) If the test signals, what is the probability that chlorinated compounds are present?

Consider the endothermic reactions in Exercise 2-50. Use BayesTheorem to calculate the probability that a reaction final temperature is 271 K or less given that the heat absorbed is above target.

Consider the hospital emergency room data in Example 2-8. Use BayesTheorem to calculate the probability that a person visits Hospital 4 given they are LWBS.

Consider the well failure data in Exercise 2-53. Use Bayes Theorem to calculate the probability that a randomly selected well is in the gneiss group given that the well is failed.

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 are 0.8 and 0.2, respectively. If the site visitor arrives from a search site, the blue and green colors are 0.4 and 0.6, respectively. The proportions of visitors from affiliates and search sites are 0.3 and 0.7, respectively. What is the probability that a visitor is from a search site given that the blue ad was viewed?

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) The volume of gasoline that is lost to evaporation during the filling of a gas tank. (d) The outside diameter of a machined shaft. (e) The number of cracks exceeding one-half inch in 10 miles of an interstate highway. (f) The weight of an injection-molded plastic part. (g) The number of molecules in a sample of gas. (h) The concentration of output from a reactor. (i) The current in an electronic circuit.

Samples of laboratory glass are in small, light packaging or heavy, large packaging. Suppose that 2 and 1% of the sample shipped in small and large packages, respectively, break during transit. If 60% of the samples are shipped in large packages and 40% are shipped in small packages, what proportion of samples break during shipment?

A sample of three calculators is selected from a manufacturing line, and each calculator is classified as either defective or acceptable. Let A, B, and C denote the events that the first, second, and third calculators, respectively, are defective. (a) Describe the sample space for this experiment with a tree diagram. Use the tree diagram to describe each of the following events: (b) A (c) B (d) (e)

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: edge finish excellent good surface excellent 80 2 finish good 10 8 Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent edge finish. If a part is selected at random, determine the following probabilities: (a) (b) (c) (d) (e) (f)

Shafts are classified in terms of the machine tool that was used for manufacturing the shaft and conformance to surface finish and roundness.

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? Tool 1 roundness conforms yes no surface finish yes 200 1 conforms no 4 2 Tool 2 roundness conforms yes no surface finish yes 145 4 conforms no 8 6 (a) If a shaft is selected at random, what is the probability that the shaft conforms to surface finish requirements or to roundness requirements or is from Tool 1? (b) If a shaft is selected at random, what is the probability that the shaft conforms to surface finish requirements or does not conform to roundness requirements or is from Tool 2? (c) If a shaft is selected at random, what is the probability that the shaft conforms to both surface finish and roundness requirements or the shaft is from Tool 2? (d) If a shaft is selected at random, what is the probability that the shaft conforms to surface finish requirements or the shaft is from Tool 2? 2

The analysis of shafts for a compressor is summarized by conformance to specifications: roundness conforms yes no surface finish yes 345 5 conforms no 12 8 (a) If we know that a shaft conforms to roundness requirements, what is the probability that it conforms to surface finish requirements? (b) If we know that a shaft does not conform to roundness requirements, what is the probability that it conforms to surface finish requirements?

A researcher receives 100 containers of oxygen. Of those containers, have oxygen that is not ionized, and the rest are ionized. Two samples are randomly selected, without replacement, from the lot. (a) What is the probability that the first one selected is not ionized? (b) What is the probability that the second one selected is not ionized given that the first one was ionized? (c) What is the probability that both are ionized? (d) How does the answer in part (b) change if samples seP1A B2 P1A B2 lected were replaced prior to the next selection?

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, and let B denote the event that the second casting is selected from the local supplier. Determine: (a) (b) (c) (d) Suppose three castings are selected at random, without replacement, from the lot of 40. In addition to the definitions of events A and B, let C denote the event that the third casting selected is from the local supplier. Determine: (e)

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 material from a single lot require reprocessing. However, the viscosity of batches consisting of two or more lots of material is more difficult to control, and 40% of such batches require additional processing to achieve the required viscosity. Let A denote the event that a batch is formed from two different lots, and let B denote the event that a lot requires additional processing. Determine the following probabilities: (a) (b) (c) (d) (e) (f) (g)

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 installation; in the remaining 3% of complaints the user has improperly followed the installation instructions. The requests for information are evenly divided on technical questions (50%) and requests to purchase more products (50%). (a) What is the probability that an incoming call to the customer service center will be from a customer who has not followed installation instructions properly? (b) Find the probability that an incoming call is a request for purchasing more products.

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 any to be resent? (b) What is the probability that an e-mail message with 3 packets will need exactly one to be resent? (c) If 10 e-mail messages are sent, each with 100 packets, what is the probability that at least one message will need some packets to be resent?

Samples of a cast aluminum part are classified on the basis of surface finish (in microinches) and length measurements. The results of 100 parts are summarized as follows: length excellent good surface excellent 80 2 finish good 10 8 Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent length. Are events A and B independent?

An optical 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 disk is eliminated as unacceptable for data storage. On an acceptable portion of the disk, the probability of a satisfactory readback is 0.98. Assume the readbacks are independent. What is the probability that an acceptable portion of the disk is eliminated as unacceptable for data storage?

Semiconductor lasers used in optical storage products require higher power levels for write operations than for read operations. High-power-level operations lower the useful life of the laser. Lasers in products used for backup of higher-speed magnetic disks primarily write, and the probability that the useful life exceeds five years is 0.95. Lasers that are in products that are used for main storage spend approximately an equal amount of time reading and writing, and the probability that the useful life exceeds five years is 0.995. Now, 25% of the products from a manufacturer are used for backup and 75% of the products are used for main storage. Let A denote the event that a lasers useful life exceeds five years, and let B denote the event that a laser is in a product that is used for backup. Use a tree diagram to determine the following: (a) (b) (c) (d) (e) (f) (g) What is the probability that the useful life of a laser exceeds five years? (h) What is the probability that a laser that failed before five years came from a product used for backup?

Energy released from three 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 only sustain maximal muscle power for less than five seconds (a short dash). Three systems are used to replenish ATP: phosphagen system; glycogen-lactic acid system (anaerobic); and aerobic respiration, but the first is only useful for less than 10 seconds, and even the second system provides less than two minutes of ATP. An endurance athlete needs to perform below the anaerobic threshold to sustain energy for extended periods. A sample of 100 individuals is described by the energy system used in exercise at different intensity levels. primarily aerobic period yes no 1 50 7 2 13 30 Let A denote the event that an individual is in period 2, and let B denote the event that the energy is primarily aerobic. Determine the number of individuals in (a) (b) (c)

A sample preparation for a chemical measurement is completed correctly by 25% of the lab technicians, completed with a minor error by 70%, and completed with a major error by 5%. (a) If a technician is selected randomly to complete the preparation, what is the probability it is completed without error? (b) What is the probability t

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 this experiment.

The data from 200 machined parts are summarized as follows: depth of bore above below edge condition target target coarse 15 10 moderate 25 20 smooth 50 80 (a) What is the probability that a part selected has a moderate edge condition and a below-target bore depth? (b) What is the probability that a part selected has a moderate edge condition or a below-target bore depth? (c) What is the probability that a part selected does not have a moderate edge condition or does not have a below-target bore depth?

Computers in a shipment of 100 units contain a portable hard drive, DVD drive, or both, according to the following table: portable hard drive yes no DVD yes 15 80 no 4 1 Let A denote the event that a computer has a portable hard drive and let B denote the event that a computer has a DVD drive. If one computer is selected randomly, compute (a) (b) (c) (d) (e)

The probability that a customers 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 all are shipped on time? (b) What is the probability that exactly one is not shipped on time? (c) What is the probability that two or more orders are not shipped on time?

Let E1, E2, and E3 denote the samples that conform to a percentage of solids specification, a molecular weight specification, and a color specification, respectively. A total of 240 samples are classified by the E1, E2, and E3 specifications, where yes indicates that the sample conforms. E3 yes E2 yes no Total E1 yes 200 1 201 no 5 4 9 Total 205 5 210 E3 no E2 yes no E1 yes 20 4 24 no 6 0 6 Total 26 4 30 (a) Are E1, E2, and E3 mutually exclusive events? (b) Are E 1, E 2, and E 3 mutually exclusive events? (c) What is P(E 1 or E 2 or E 3)? (d) What is the probability that a sample conforms to all three specifications? (e) What is the probability that a sample conforms to the E1 or E3 specification? (f) What is the probability that a sample conforms to the E1 or E2 or E3 specification?

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 time. If 30% of transactions are changes, what is the probability that a transaction can be completed in less than 100 milliseconds?

A steel plate contains 20 bolts. Assume that five bolts are not torqued to the proper limit. Four bolts are selected at random, without replacement, to be checked for torque. (a) What is the probability that all four of the selected bolts are torqued to the proper limit? (b) What is the probability that at least one of the selected bolts is not torqued to the proper limit?

The following circuit operates if and only if there is a path of functional devices from left to right. Assume devices fail independently and that the probability of failure of each device is as shown. What is the probability that the circuit operates?

The probability of getting through by telephone to buy concert tickets is 0.92. For the same event, the probability of accessing the vendors Web site is 0.95. Assume that these two ways to buy tickets are independent. What is the probability that someone who tries to buy tickets through the Internet and by phone will obtain tickets?

The British government has stepped up its information campaign regarding foot-and-mouth disease by mailing brochures to farmers around the country. It is estimated that 99% of Scottish farmers who receive the brochure possess enough information to deal with an outbreak of the disease, but only 90% of those without the brochure can deal with an outbreak. After the first three months of mailing, 95% of the farmers in Scotland received the informative brochure. Compute the probability that a randomly selected farmer will have enough information to deal effectively with an outbreak of the disease

In an automated filling operation, the probability of an incorrect fill when the process is operated at a low speed is 0.001. When the process is operated at a high speed, the probability of an incorrect fill is 0.01. Assume that 30% of the containers are filled when the process is operated at a high speed and the remainder are filled when the process is operated at a low speed. (a) What is the probability of an incorrectly filled container? (b) If an incorrectly filled container is found, what is the probability that it was filled during the high-speed operation?

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 the errors are independent. (a) What is the probability of a completely defect-free message? (b) What is the probability of a message that has either an encode or a decode error?

It is known that two defective copies of a commercial software program were erroneously sent to a shipping lot that now has a total of 75 copies of the program. A sample of copies will be selected from the lot without replacement.(a) If three copies of the software are inspected, determine the probability that exactly one of the defective copies will be found. (b) If three copies of the software are inspected, determine the probability that both defective copies will be found. (c) If 73 copies are inspected, determine the probability that both copies will be found. Hint: Work with the copies that remain in the lot.

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 that the tool fails during the warranty period?

An e-mail message can travel through one of two server routes. The probability of transmission error in each of the servers and the proportion of messages that travel each route are shown in the following table. Assume that the servers are independent. probability of error percentage of messages server 1 server 2 server 3 server 4 route 1 30 0.01 0.015 route 2 70 0.02 0.003 (a) What is the probability that a message will arrive without error? (b) If a message arrives in error, what is the probability it was sent through route 1?

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 requests? (b) What is the probability that the machine is idle at the time of exactly four of your requests? (c) What is the probability that the tool is idle at the time of at least three of your requests?

A lot of 50 spacing washers contains 30 washers that are thicker than the target dimension. Suppose that three washers are selected at random, without replacement, from the lot. (a) What is the probability that all three washers are thicker than the target? (b) What is the probability that the third washer selected is thicker than the target if the first two washers selected are thinner than the target? (c) What is the probability that the third washer selected is thicker than the target?

Continuation of Exercise 2-187. Washers are selected from the lot at random, without replacement.(a) What is the minimum number of washers that need to be selected so that the probability that all the washers are thinner than the target is less than 0.10? (b) What is the minimum number of washers that need to be selected so that the probability that one or more washers are thicker than the target is at least 0.90?

The following table lists the history of 940 orders for features in an entry-level computer product. extra memory no yes optional high- no 514 68 speed processor yes 112 246 Let A be the event that an order requests the optional highspeed processor, and let B be the event that an order requests extra memory. Determine the following probabilities: (a) (b) (c) (d) (e) What is the probability that an order requests an optional high-speed processor given that the order requests extra memory? (f) What is the probability that an order requests extra memory given that the order requests an optional high-speed processor?

The alignment between the magnetic tape and head in a magnetic tape storage system affects the performance of the system. Suppose that 10% of the read operations are degraded by skewed alignments, 5% of the read operations are degraded by off-center alignments, and the remaining read operations are properly aligned. The probability of a read error is 0.01 from a skewed alignment, 0.02 from an off-center alignment, and 0.001 from a proper alignment. (a) What is the probability of a read error? (b) If a read error occurs, what is the probability that it is due to a skewed alignment?

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 not operate?

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 provide contact information. Use the following tables to answer the questions: Number of pages viewed: 1 2 3 4 or more Percentage of visitors: 40 30 20 10 Percentage of visitors in each page-view category that provide contact information: 10 10 20 40 (a) What is the probability that a visitor to the Web site provides contact information? (b) If a visitor provides contact information, what is the probability that the visitor viewed four or more pages?

An article in Genome Research [An Assessment of Gene Prediction Accuracy in Large DNA Sequences (2000, Vol. 10, pp. 16311642)], considered the accuracy of commercial software to predict nucleotides in gene sequences. The following table shows the number of sequences for which the programs produced predictions and the number of nucleotides correctly predicted (computed globally from the total number of prediction successes and failures on all sequences). Number of Sequences Proportion GenScan 177 0.93 Blastx default 175 0.91 Blastx topcomboN 174 0.97 Blastx 2 stages 175 0.90 GeneWise 177 0.98 Procrustes 177 0.93 Assume the prediction successes and failures are independent among the programs. (a) What is the probability that all programs predict a nucleotide correctly? (b) What is the probability that all programs predict a nucleotide incorrectly? (c) What is the probability that at least one Blastx program predicts a nucleotide correctly?

A batch contains 36 bacteria cells. Assume that 12 of the cells are not capable of cellular replication. Six cells are selected at random, without replacement, to be checked for replication. (a) What is the probability that all six cells of the selected cells are able to replicate? (b) What is the probability that at least one of the selected cells is not capable of replication?

A computer system uses passwords that are exactly seven characters and each character is one of the 26 letters (az) or 10 integers (09). Uppercase letters are not used. (a) How many passwords are possible? (b) If a password consists of exactly six letters and one number, how many passwords are possible? (c) If a password consists of five letters followed by two numbers, how many passwords are possible?

Natural red hair consists of two genes. People with red hair have two dominant genes, two regressive genes, or one dominant and one regressive gene. A group of 1000 people was categorized as follows: Gene 2 Gene 1 Dominant Regressive Other Dominant 5 25 30 Regressive 7 63 35 Other 20 15 800 Let A denote the event that a person has a dominant red hair gene and let B denote the event that a person has a regressive red hair gene. If a person is selected at random from this group, compute the following. (a) (b) (c) (d) (e) (f ) Probability that the selected person has red hair

Two suppliers each supplied 2000 parts and these were evaluated for conformance to specifications. One part type was of greater complexity than the other. The proportion of nonconforming parts of each type are shown in the table. Simple Complex Supplier Component Assembly Total 1 Nonconforming 2 10 12 Total 1000 1000 2000 2 Nonconforming 4 6 10 Total 1600 400 2000 One part is selected at random from each supplier. For each supplier, separately calculate the following probabilities: (a) What is the probability a part conforms to specifications? (b) What is the probability a part conforms to specifications given it is a complex assembly? (c) What is the probability a part conforms to specifications given it is a simple component? (d) Compare your answers for each supplier in part (a) to those in parts (b) and (c) and explain any unusual results

Suppose documents in a lending organization are selected randomly (without replacement) for review. In a set of 50 documents, suppose that two actually contain errors. (a) What is the minimum sample size such that the probability exceeds 0.90 that at least one document in error is selected? (b) Comment on the effectiveness of sampling inspection to detect errors

Suppose that a lot of washers is large enough that it can be assumed that the sampling is done with replacement. Assume that 60% of the washers exceed the target thickness. (a) What is the minimum number of washers that need to be selected so that the probability that none are thicker than the target is less than 0.10? (b) What is the minimum number of washers that need to be selected so that the probability that one or more washers are thicker than the target is at least 0.90?

A biotechnology manufacturing firm can produce diagnostic test kits at a cost of $20. Each kit for which there is a demand in the week of production can be sold for $100. However, the half-life of components in the kit requires the kit to be scrapped if it is not sold in the week of production. The cost of scrapping the kit is $5. The weekly demand is summarized as follows: weekly demand Number of units 0 50 100 200 Probability of demand 0.05 0.4 0.3 0.25 How many kits should be produced each week to maximize the mean earnings of the firm?

Assume the following characteristics of the inspection process in Exercise 2-177. If an operator checks a bolt, the probability that an incorrectly torqued bolt is identified is 0.95. If a checked bolt is correctly torqued, the operators conclusion is always correct. What is the probability that at least one bolt in the sample of four is identified as being incorrectly torqued?

If the events A and B are independent, show that and are independent.

Suppose that a table of part counts is generalized as follows: conforms yes no supplier 1 ka kb 2 a b where a, b, and k are positive integers. Let A denote the event that a part is from supplier 1 and let B denote the event that a part conforms to specifications. Show that A and B are independent events. This exercise illustrates the result that whenever the rows of a table (with r rows and c columns) are proportional, an event defined by a row category and an event defined by a column category are independent.