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The number of hits (views) is recorded at a high-volume Web site in a day.

Provide a reasonable description of the sample space. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.


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

Provide a reasonable description of the sample space for each of the random experiments. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.

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

Provide a reasonable description of the sample space for each of the random experiments. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.

The concentration of ozone to the nearest part per billion.

Provide a reasonable description of the sample space for each of the random experiments. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.

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

Provide a reasonable description of the sample space for each of the random experiments. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.

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

Provide a reasonable description of the sample space for each of the random experiments. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.

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.

Provide a reasonable description of the sample space for each of the random experiments. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.

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

Provide a reasonable description of the sample space for each of the random experiments. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.

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.

Provide a reasonable description of the sample space for each of the random experiments. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.

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

Provide a reasonable description of the sample space for each of the random experiments. There can be more than one acceptable interpretation of each experiment. Describe any assumptions you make.

In light-dependent photosynthesis, light quality refers to the wavelengths of light that are important. The wavelength of a sample of photo synthetically active radiations (PAR) is measured to the nearest nanometer. The red range is 675–700 nm and the blue range is 450–500 nm. Let A denote the event that PAR occurs in the red range, and let B denote the event that PAR occurs in the blue range. Describe the sample space and indicate each of the following events:

(a) A

(b) B

(c) A Ç B

(d) A È B

In control replication, cells are replicated over a period of two days. Not until mitosis is completed can freshly synthesized DNA be replicated again. Two control mechanisms have been identified—one positive and one negative. Suppose that a replication is observed in three cells. Let A denote the event that all cells are identified as positive, and let B denote the event that all cells are negative. Describe the sample space graphically and display each of the following events:

(a) A

(b) B

(c) A Ç B

(d) A È B

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?

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?

The following table summarizes 204 endothermic reactions involving sodium bicarbonate.

Let A denote the event that a reaction’s final temperature is 271 K or less. Let B denote the event that the heat absorbed is below target. Determine the number of reactions in each of the following events.

(a) A Ã‡ B

(b) A¢

(c) A Ãˆ B

(d) A Ãˆ B¢

(e) A¢ Ç B¢

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 given below. Let A denote the event that a visit is to hospital 1, and let B denote the event that a visit results in admittance to any hospital. 


Determine the number of persons in each of the following events.

(a) A Ç B

(b) A¢

(c) A È B

(d) A È B¢

(e) A¢Ç B¢

An article in The Journal of Data Science [“A Statistical Analysis of Well Failures in Baltimore County” (2009, Vol. 7, pp. 111–127)] provided the following table of well failures for different geological formation groups in Baltimore County.


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) A Ç B

(b) A¢

(c) A È B

(d) A Ç B¢

(e) A¢ Ç B¢

A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. 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 shoulder surgeries scheduled consecutively?

(c) How many different schedules begin and end with a knee surgery?

Consider the bar code in Example 2-12. One code is still held back as a delimiter. For each of the following cases, how many characters can be encoded?

(a) The constraint of exactly two wide bars is replaced with one that requires exactly one wide bar.

(b) The constraint of exactly two wide bars is replaced with one that allows either one or two wide bars.

(c) The constraint of exactly two wide bars is dropped.

(d) The constraints of exactly two wide bars and one wide space are dropped.

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 the events that consist of passwords with only letters or only integers, respectively. Determine the number of passwords in each of the following events.

(a) Ω

(b) A

(c) A¢Ç B¢

(d) Passwords that contain at least 1 integer

(e) Passwords that contain exactly 1 integer

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 an 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 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 and 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?

Magnesium alkyls are used as homogenous catalysts in the production of linear low-density polyethylene (LLDPE), which requires a finer magnesium powder to sustain a reaction. Redox reaction experiments using four different amounts of magnesium powder are performed. Each result may or may not be further reduced in a second step using three different magnesium powder amounts. Each of these results may or may not be further reduced in a third step using three different amounts of magnesium powder.

(a) How many experiments are possible?

(b) If all outcomes are equally likely, what is the probability that the best result is obtained from an experiment that uses all three steps?

(c) Does the result in part (b) change if five or six or seven different amounts are used in the first step? Explain.

An article in the Journal of Database Management [“Experimental Study of a Self-Tuning Algorithm for DBMS Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload used in the TPC-C OLTP (Transaction Processing Performance Council’s Version C On-Line Transaction Processing) benchmark, which simulates a typical order entry application. 

The frequency of each type of transaction (in the second column) can be used as the percentage of each type of transaction. The average number of selects operations required for each type of transaction is shown. Let A denote the event of transactions with an average number of selects operations of 12 or fewer. Let B denote the event of transactions with an average number of updates operations of 12 or fewer. Calculate the following probabilities.

(a) P(A)

(b) P(B)

(c) P(AÇB)

(d) P(AÇB’)

(e)P(AÈB)

Consider the endothermic reaction’s table given below. 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 above target.

Determine the following probabilities.

(a) P(AÇB)

(b) P(A')

(c)P(AÈB)

(d) P(AÈB')

(e) P(A'ÇB')

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 is given below. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any hospital).

Determine the following probabilities.

(a) P(AÇB)

(b) P(A¢)

(c) P(AÈB)

(d) P(AÈB¢)

(e) P(A¢ÇB¢)

Consider the well failure data is given below. 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(AÇB)

(b) P(A¢)

(c) P(AÈB)

(d) P(AÈB¢)

(e) P(A¢ÇB¢)

Consider the bar code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (five bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a (white) space appears between each bar. The original specification (since revised) used exactly two wide bars and one wide space in each character. For example, if b and B denote narrow and wide (black) bars, respectively, and w and W denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally likely (none is held back as a delimiter).

Determine the probability for each of the following:

(a) A wide space occurs before a narrow space.

(b) Two wide bars occur consecutively.

(c) Two consecutive wide bars are at the start or end.

(d) The middle bar is wide.

A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. Suppose that an operating room needs to schedule three knee, four hip, and five shoulder surgeries. Assume that all schedules are equally likely.

Determine the probability for each of the following:

(a) All hip surgeries are completed before another type of surgery.

(b) The schedule begins with a hip surgery.

(c) The first and last surgeries are hip surgeries.

(d) The first two surgeries are hip surgeries.

Suppose that a patient is selected randomly from the those described ,The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of Chronic Hepatitis C,” [Gastroenterology (1996, Vol. 111, no. 5, pp. 1307–1312)], considered the effect of two treatments and a control for treatment of hepatitis C. The following table provides the total patients in each group and the number that showed a complete (positive) response after 24 weeks of treatment.

Let A denote the event that the patient is in the group treated with interferon alfa, and let B denote the event that the patient has a complete response.

Determine the following probabilities.

(a) P(A)

(b) P(B)

(c)P(AÇB)

(d) P(AÈB)

(e) P(A¢ÈB)

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 events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in Ω are equally likely.

Determine the probability of each of the following:

(a) A

(b) B

(c) A password contains at least 1 integer.

(d) A password contains exactly 2 integers.

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 following causes for knee injuries were considered:

Activity                                               Percentage of Knee injuries

Contact sport…………………………………………………..46%

Noncontact sport………………………………………………44%

Activity of daily living………………………………………….9%

Riding motorcycle………………………………………………1%

(a) What is the probability that a knee injury resulted from a sport (contact or noncontact)?

(b) What is the probability that a knee injury resulted from an activity other than a sport?

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 a strand is randomly selected, what is the probability that its conductivity is low or its strength is low?

(c) Consider the event that a strand has low conductivity and the event that the strand has low strength. Are these two events mutually exclusive?

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 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) P(A)

(b) P(B)

(c) P(AÇB)

(d) P(AÈB)

Consider the endothermic reactions given below. 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 above target. 

Use the addition rules to calculate the following probabilities.

(a) P(AÈB)

(b) P(AÇB¢)

(c) P(A¢ÈB¢)

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) P(AÈB)

(b) P(AÈB¢)

(c) P(A¢ÈB¢)

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.

Consider the code 39 is a common bar code system that consists of narrow and wide bars(black) separated by either wide or narrow spaces (white). Each character contains nine elements (five bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a (white) space appears between each bar. The original specification (since revised) used exactly two wide bars and one wide space in each character. For example, if b and B denote narrow and wide (black) bars, respectively, and w and W denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally likely (none is held back as a delimiter).

Let A and B denote the event that the first bar is wide and B denote the event that the second bar is wide.

Determine the following:

(a) P(A)

(b) P(B)

(c) P(A Ç B)

(d) Are A and B independent events?

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 the value for p so that the probability that the integrated circuit functions is 0.95.

The following table 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?



The following table provides data on wafers categorized by location and contamination levels. More generally, let the number of wafers with low contamination from the center and edge locations be denoted as nlcand nle, respectively. Similarly, let nhcand nhedenote the number of wafers with high contamination from the center and edge locations, respectively. Suppose that nlc= 10nhcand nle= 10nhe. That is, there are 10 times as many low contamination wafers as high ones from each location. 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? Does your conclusion change if the multiplier of 10 (between low and high contamination wafers) is changed from10 to another positive integer?

Consider the endothermic reactions given below. Use Bayes’ theorem to calculate the probability that a reaction's final temperature is 271 K or less given that the heat absorbed is above target.


Consider the hospital emergency room data given below. Use Bayes’ theorem to calculate the probability that a person visits hospital 4 given they are LWBS.


Consider the well failure data given below. Use Bayes’ theorem to calculate the probability that a randomly selected well is in the gneiss group given that the well has 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 being used in the advertisement are 0.8 and 0.2, respectively. If the site visitor arrives from a search site, the probabilities of blue and green colors in the advertisement 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?

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 with different drug therapies (such as prednisone and infliximab): sequential monotherapy (group 1), step-up combination therapy (group 2), initial combination therapy (group 3), or initial combination therapy with infliximab (group 4). Radiographs of hands and feet were used to evaluate disease progression. The number of patients without progression of joint damage was 76 of 114 patients (67%), 82 of 112 patients (73%), 104 of 120 patients (87%), and 113 of 121 patients (93%) in groups 1–4, respectively. Suppose that a patient is selected randomly. 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 | A)

(c) P (A | B)

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.

(b) The message is spam given that it contains free.

(c) The message is valid given that it does not contain free.

The probabilities of poor print quality given no printer problem, misaligned paper, high ink viscosity, or printer-head debris are 0, 0.3, 0.4, and 0.6, respectively. The probabilities of no printer problem, misaligned paper, high ink viscosity, or printer-head debris are 0.8, 0.02, 0.08, and 0.1, respectively.

(a) Determine the probability of high ink viscosity given poor print quality.

(b) Given poor print quality, what problem is most likely?

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

(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?

A researcher receives 100 containers of oxygen. Of those containers, 20 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 selected were replaced prior to the next selection?

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 the probability that an e-mail message with 3 packets will need exactly 1 to be resent?

(c) If 10 e-mail messages are sent, each with 100 packets, what is the probability that at least 1 message will need some packets to be resent?

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 short dash). Three systems are used to replenish ATP—phosphagen system, glycogen-lactic acid system (anaerobic), and aerobic respiration—but the first is useful only 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.


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) A'Ç B

(b) B'

(c) A È B

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 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).

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. 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 cells are able to replicate?

(b) What is the probability that at least 1 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 (a–z) or 10 integers (0–9). Uppercase letters are not used.

(a) How many passwords are possible?

(b) If a password consists of exactly 6 letters and 1 number, how many passwords are possible?

(c) If a password consists of 5 letters followed by 2 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:


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) P(A)

(b) P(A ∩ B)

(c) P(A∪ B)

(d) P(A ∩ B)

(e) P(A | B)

(f) Probability that the selected person has red hair

Two suppliers each supplied 2000 parts that were evaluated for conformance to specifications. One part type was more complex than the other. The proportion of nonconforming parts of each type are shown in the table.

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.

An article in Knee Surgery, Sports Traumatology, Arthroscopy [“Arthroscopic Meniscal Repair with an Absorbable Screw: Results and Surgical Technique” (2005, Vol. 13, pp. 273–279)] cites a success rate of more than 90% for meniscal tears with a rim width under 3 mm, but only a 67% success rate for tears of 3–6 mm. If you are unlucky enough to suffer a meniscal tear of under 3 mm on your left knee and one of width 3–6 mm on your right knee, what is the probability mass function of the number of successful surgeries? Assume that the surgeries are independent.

The space shuttle flight control system called Primary Avionics Software Set (PASS) uses four independent computers working in parallel. At each critical step, the computers “vote” to determine the appropriate step. The probability that a computer will ask for a roll to the left when a roll to the right is appropriate is 0.0001. Let X denote the number of computers that vote for a left roll when a right roll is appropriate. What is the probability mass function of X?

The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows:

Final Temperature Conditions                    Number of Reactions

266 K……………………………………………………………..48

271 K……………………………………………………………..60

274 K……………………………………………………………..92

Calculate the probability mass function of final temperature.

Actual lengths of stay at a hospital’s emergency department in 2009 are shown in the following table (rounded to the nearest hour). Length of stay is the total of wait and service times. Some longer stays are also approximated as 15 hours in this table.

Calculate the probability mass function of the wait time for service.

The distribution of the time until a Web site changes is important to Web crawlers that search engines use to maintain current information about Web sites. The distribution of the time until change (in days) of a Web site is approximated in the following table.

Days until Changes               Probability

1.5…………………………………...0.05

3.0…………………………………...0.25

4.5…………………………………...0.35

5.0…………………………………...0.20

7.0…………………………………...0.15

Calculate the probability mass function of the days until change.

The following table shows the typical depth (rounded to the nearest foot) for nonfailed wells in geological formations in Baltimore County (The Journal of Data Science, 2009, Vol. 7, pp. 111–127).

Calculate the probability mass function of depth for nonfailed wells from the table.

Consider the wafers with contamination particles in Example 2-17. Assume that wafers are independent with respect to contamination particles. Wafers are selected until one with five or more contamination particles occurs. What is the probability mass function of the number of wafers selected?

Consider the circuit in Example 2-32. Assume that devices fail independently. What is the probability mass function of the number of failed devices?

In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states. Assume the following proportions of the states:

Nickel Charge                        Proportions Found

0……………………………………………...0.17

+2…………………………………………….0.35

+3…………………………………………….0.33

+4…………………………………………….0.15

(a) Determine the cumulative distribution function of nickel charge.

(b) Determine the mean and variance of the nickel charge.

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 Transaction Processing Performance Council’s Version C On-Line Transaction Processing (TPC-C OLTP) benchmark, which simulates a typical order entry application.

The frequency of each type of transaction (in the second column) can be used as the percentage of each type of transaction. The average number of selects operations required for each type of transaction is shown.

(a) Determine the mean and standard deviation of the number of selects operations for a transaction from the distribution of types shown in the table.

(b) Determine the mean and standard deviation of the total number of operations (selects, updates,…, and joins) for a transaction from the distribution of types shown in the table.

Let the random variable X have a discrete uniform distribution on the integers 0 ≤ x ≤ 99. Determine the mean and variance of X.

Let the random variable X have a discrete uniform distribution on the integers 1 ≤ x ≤ 3. Determine the mean and variance of X.

Assume that the wavelengths of photosynthetically active radiations (PAR) are uniformly distributed at integer nanometers in the red spectrum from 675 to 700 nm.

(a) What are the mean and variance of the wavelength distribution for this radiation?

(b) If the wavelengths are uniformly distributed at integer nanometers from 75 to 100 nanometers, how do the mean and variance of the wavelength distribution compare to the previous part? Explain.

The number of pages in a PDF document you create has a discrete uniform distribution from five to nine pages (including the end points). What are the mean and standard deviation of the number of pages in the document?

Suppose that nine-digit Social Security numbers are assigned at random. If you randomly select a number, what is the probability that it belongs to one of the 300 million people in the United States?

Suppose that 1000 seven-digit telephone numbers within your area code are dialed randomly. What is the probability that your number is called?

The probability that data are entered incorrectly into a field in a database is 0.005. A data entry form has 28 fields, and errors occur independently for each field. The random variable X is the number of fields on the form with an error. Does X have a discrete uniform distribution? Why or why not?

Each multiple-choice question on an exam has four choices. Suppose that there are 10 questions and the choice is selected randomly and independently for each question. Let X denote the number of questions answered correctly. Does X have a discrete uniform distribution? Why or why not?

Consider the hospital data in Example 2-8. Suppose a patient is selected randomly from the collection in the table. Let X denote the hospital number of the selected patient (either 1, 2, 3, or 4). Does X have a discrete uniform distribution? Why or why not?

Samples of rejuvenated mitochondria are mutated (defective) in 1% of cases. Suppose that 15 samples are studied and can be considered to be independent for mutation. Determine the following probabilities. The binomial table in Appendix A can help.

(a) No samples are mutated.

(b) At most one sample is mutated.

(c) More than half the samples are mutated.

An article in Information Security Technical Report [“Malicious Software—Past, Present and Future” (2004, Vol. 9, pp. 6–18)] provided the following data on the top 10 malicious software instances for 2002. The clear leader in the number of registered incidences for the year 2002 was the Internet worm “Klez,” and it is still one of the most widespread threats. This virus was first detected on 26 October 2001, and it has held the top spot among malicious software for the longest period in the history of virology.

The 10 most widespread malicious programs for 2002

Suppose that 20 malicious software instances are reported.

Assume that the malicious sources can be assumed to be independent.

(a) What is the probability that at least one instance is “Klez?”

(b) What is the probability that three or more instances are “Klez?”

(c) What are the mean and standard deviation of the number of “Klez” instances among the 20 reported?

Consider the lengths of stay at a hospital’s emergency department in Exercise 3-33. Assume that five persons independently arrive for service.

(a) What is the probability that the length of stay of exactly one person is less than or equal to 4 hours?

(b) What is the probability that exactly two people wait more than 4 hours?

(c) What is the probability that at least one person waits more than 4 hours?

Consider the patient data in Example 2-8. Suppose that five patients are randomly selected with replacement from the total for hospital 4. Determine the following probabilities:

(a) Exactly one is LWBS.

(b) Two or more are LWBS.

(c) At least one is LWBS.

Assume that a Web site changes its content according to the distribution in Exercise 3-34. Assume that 10 changes are made independently.

(a) What is the probability that the change is made in less than 4 days in 7 of the 10 updates?

(b) What is the probability that the change is made in less than 4 days in 2 or fewer of the 10 updates?

(c) What is the probability that at least one change is made in less than 4 days?

(d) What is the expected number of the 10 updates that occur in less than 4 days?

Consider the endothermic reactions in Exercise 3-32. A total of 20 independent reactions are to be conducted.

(a) What is the probability that exactly 12 reactions result in a final temperature less than 272 K?

(b) What is the probability that at least 19 reactions result in a final temperature less than 272 K?

(c) What is the probability that at least 18 reactions result in a final temperature less than 272 K?

(d) What is the expected number of reactions that result in a final temperature of less than 272 K?

The probability that a visitor to a Web site provides contact data for additional information is 0.01. Assume that 1000 visitors to the site behave independently. Determine the following probabilities:

(a) No visitor provides contact data.

(b) Exactly 10 visitors provide contact data.

(c) More than 3 visitors provide contact data.

Consider the circuit in Example 2-34. Assume that devices fail independently. What is the probability mass function of the number of device failures? Explain why a binomial distribution does not apply to the number of device failures in Example 2-32.

Consider the time to recharge the flash in cell-phone cameras as in Example 3-2. Assume that the probability that a camera passes the test is 0.8 and the cameras perform independently. What is the smallest sample size needed so that the probability of at least one camera failing is at least 95%?

Consider the patient data in Example 2-8. Suppose that patients are randomly selected with replacement from the total for hospital 4. What is the smallest sample size needed so that the probability is at least 90% that at least one patient is LWBS?

A computer system uses passwords constructed from the 26 letters (a–z) or 10 integers (0–9). Suppose that 10,000 users of the system have unique passwords. A hacker randomly selects (with replacement) passwords from the potential set.

(a) Suppose that 9900 users have unique six-character passwords and the hacker randomly selects six-character passwords. What are the mean and standard deviation of the number of attempts before the hacker selects a user password?

(b) Suppose that 100 users have unique three-character passwords and the hacker randomly selects three-character passwords.

What are the mean and standard deviation of the number of attempts before the hacker selects a user password?

(c) Comment on the security differences between six- and three-character passwords.

In the process of meiosis, a single parent diploid cell goes through eight different phases. However, only 60% of the processes pass the first six phases and only 40% pass all eight. Assume that the results from each phase are independent.

(a) If the probability of a successful pass of each one of the first six phases is constant, what is the probability of a successful pass of a single one of these phases?

(b) If the probability of a successful pass of each one of the last two phases is constant, what is the probability of a successful pass of a single one of these phases?

A Web site randomly selects among 10 products to discount each day. The color printer of interest to you is discounted today.

(a) What is the expected number of days until this product is again discounted?

(b) What is the probability that this product is first discounted again exactly 10 days from now?

(c) If the product is not discounted for the next five days, what is the probability that it is first discounted again 15 days from now?

(d) What is the probability that this product is first discounted again within three or fewer days?

Consider the visits that result in leave without being seen (LWBS) at an emergency department in Example 2-8. Assume that people independently arrive for service at hospital l.

(a) What is the probability that the fifth visit is the first one to LWBS?

(b) What is the probability that either the fifth or sixth visit is the first one to LWBS?

(c) What is the probability that the first visit to LWBS is among the first four visits?

(d) What is the expected number of visits until the third LWBS occurs?

An array of 30 LED bulbs is used in an automotive light. The probability that a bulb is defective is 0.001 and defective bulbs occur independently. Determine the following:

(a) Probability that an automotive light has two or more defective bulbs.

(b) Expected number of automotive lights to check to obtain one with two or more defective bulbs.

Consider the patient data in Example 2-8. Suppose that patients are randomly selected with replacement, from the total for hospital 4. Determine the following:

(a) Probability that the first patient admitted is the first one selected.

(b) Probability that four or fewer patients are selected to admit two.

(c) Expected number of patients selected to admit 10.

Customers visit a Web site, and the probability of an order if a customer views five or fewer pages is 0.01. However, if a customer views more than five pages, the probability of an order is 0.1. The probability a customer views five or more pages is 0.25. The customers behave independently.

(a) Is the number of customers who visit the site until an order is obtained a geometric random variable? Why or why not?

(b) What is the probability that the first order is obtained from the tenth customer to visit the site?

The analysis of results from a leaf transmutation experiment (turning a leaf into a petal) is summarized by the type of transformation completed:

A naturalist randomly selects three leaves from this set without replacement. Determine the following probabilities.

(a) Exactly one has undergone both types of transformations.

(b) At least one has undergone both transformations.

(c) Exactly one has undergone one but not both transformations.

(d) At least one has undergone at least one transformation.

Consider the visits that result in leave without being seen (LWBS) at an emergency department in Example 2-8. Assume that four visits that result in LWBS are to be randomly selected (without replacement) for a follow-up interview.

(a) What is the probability that all selected visits are from hospital 4?

(b) What is the probability that no selected visits are from hospital 4?

(c) What is the probability that all selected visits are from the same hospital?

Consider the nonfailed wells in Exercises 3-35. Assume that four wells are selected randomly (without replacement) for inspection.

(a) What is the probability that exactly two are selected from the Loch Raven Schist?

(b) What is the probability that one or more is selected from the Loch Raven Schist?

(c) What is the expected number selected from the Loch Raven Schist?

Consider the semiconductor wafer data in Table 2-1. Suppose that 10 wafers are selected randomly (without replacement) for an electrical test. Determine the following:

(a) Probability that exactly 4 wafers have high contamination.

(b) Probability that at least 1 is from the center of the sputtering tool and has high contamination.

(c) Probability that exactly 3 have high contamination or are from the edge of the sputtering tool.

(d) Instead of 10 wafers, what is the minimum number of wafers that need to be selected so that the probability that at least 1 wafer has high contamination is greater than or equal to 0.9?

Suppose that a healthcare provider selects 20 patients randomly (without replacement) from among 500 to evaluate adherence to a medication schedule. Suppose that 10% of the 500 patients fail to adhere with the schedule. Determine the following:

(a) Probability that exactly 10% of the patients in the sample fail to adhere.

(b) Probability that fewer than 10% of the patients in the sample fail to adhere.

(c) Probability that more than 10% of the patients in the sample fail to adhere.

(d) Mean and variance of the number of patients in the sample who fail to adhere.

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