Solved: An inventory study determines that, on average,

Problem 60E

A random variable X that assumes the values x1,x2,...,xk is called a discrete uniform random variable if its probability mass function is f(x)=1 k for all of x1,x2,...,xk and 0 otherwise. Find the mean and variance of X.

Twelve people are given two identical speakers, which they are asked to listen to for differences, if any. Suppose that these people answer simply by guessing. Find the probability that three people claim to have heard a difference between the two speakers.

An employee is selected from a staff of 10 to supervise a certain project by selecting a tag at random from a box containing 10 tags numbered from 1 to 10. Find the formula for the probability distribution of X representing the number on the tag that is drawn. What is the probability that the number drawn is less than 4?

In a certain city district, the need for money to buy drugs is stated as the reason for 75% of all thefts. Find the probability that among the next 5 theft cases reported in this district, (a) exactly 2 resulted from the need for money to buy drugs; (b) at most 3 resulted from the need for money to buy drugs.

According to Chemical Engineering Progress (November 1990), approximately 30% of all pipework failures in chemical plants are caused by operator error. (a) What is the probability that out of the next 20 pipework failures at least 10 are due to operator error? (b) What is the probability that no more than 4 out of 20 such failures are due to operator error? (c) Suppose, for a particular plant, that out of the random sample of 20 such failures, exactly 5 are due to operator error. Do you feel that the 30% gure stated above applies to this plant? Comment.

According to a survey by the Administrative Management Society, one-half of U.S. companies give employees 4 weeks of vacation after they have been with the company for 15 years. Find the probability that among 6 companies surveyed at random, the number that give employees 4 weeks of vacation after 15 years of employment is (a) anywhere from 2 to 5; (b) fewer than 3.

One prominent physician claims that 70% of those with lung cancer are chain smokers. If his assertion is correct, (a) nd the probability that of 10 such patients recently admitted to a hospital, fewer than half are chain smokers; (b) nd the probability that of 20 such patients recently admitted to a hospital, fewer than half are chain smokers.

According to a study published by a group of University of Massachusetts sociologists, approximately 60% of the Valium users in the state of Massachusetts first took Valium for psychological problems. Find the probability that among the next 8 users from this state who are interviewed, (a) exactly 3 began taking Valium for psychological problems; (b) at least 5 began taking Valium for problems that were not psychological.

In testing a certain kind of truck tire over rugged terrain, it is found that 25% of the trucks fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability that (a) from 3 to 6 have blowouts; (b) fewer than 4 have blowouts; (c) more than 5 have blowouts.

A nationwide survey of college seniors by the University of Michigan revealed that almost 70% disapprove of daily pot smoking, according to a report in Parade. If 12 seniors are selected at random and asked their opinion, nd the probability that the number who disapprove of smoking pot daily is (a) anywhere from 7 to 9; (b) at most 5; (c) not less than 8.

The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive?

A trac control engineer reports that 75% of the vehicles passing through a checkpoint are from within the state. What is the probability that fewer than 4 of the next 9 vehicles are from out of state?

A national study that examined attitudes about antidepressants revealed that approximately 70% of respondents believe antidepressants do not really cure anything, they just cover up the real trouble. According to this study, what is the probability that at least 3 of the next 5 people selected at random will hold this opinion?

The percentage of wins for the Chicago Bulls basketball team going into the playos for the 199697 season was 87.7. Round the 87.7 to 90 in order to use Table A.1. (a) What is the probability that the Bulls sweep (4-0) the initial best-of-7 playo series? (b) What is the probability that the Bulls win the initial best-of-7 playo series? (c) What very important assumption is made in answering parts (a) and (b)?

It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, nd the probability that (a) none contracts the disease; (b) fewer than 2 contract the disease; (c) more than 3 contract the disease.

Suppose that airplane engines operate independently and fail with probability equal to 0.4. Assuming that a plane makes a safe flight if at least one-half of its engines run, determine whether a 4-engine plane or a 2- engine plane has the higher probability for a successful flight.

If X represents the number of people in Exercise 5.13 who believe that antidepressants do not cure but only cover up the real problem, find the mean and variance of X when 5 people are selected at random.

(a) In Exercise 5.9, how many of the 15 trucks would you expect to have blowouts? (b) What is the variance of the number of blowouts experienced by the 15 trucks? What does that mean?

As a student drives to school, he encounters a traffic signal. This traffic signal stays green for 35 seconds, yellow for 5 seconds, and red for 60 seconds. Assume that the student goes to school each weekday between 8:00 and 8:30 a.m. Let \(X_1\) be the number of times he encounters a green light, \(X_2\) be the number of times he encounters a yellow light, and \(X_3\) be the number of times he encounters a red light. Find the joint distribution of \(X_1\), \(X_2\), and \(X_3\).

According to USA Today (March 18, 1997), of 4 million workers in the general workforce, 5.8% tested positive for drugs. Of those testing positive, 22.5% were cocaine users and 54.4% marijuana users. (a) What is the probability that of 10 workers testing positive, 2 are cocaine users, 5 are marijuana users, and 3 are users of other drugs? (b) What is the probability that of 10 workers testing positive, all are marijuana users?(c) What is the probability that of 10 workers testing positive, none is a cocaine user?

The surface of a circular dart board has a small center circle called the bulls-eye and 20 pie-shaped regions numbered from 1 to 20. Each of the pie-shaped regions is further divided into three parts such that a person throwing a dart that lands in a specic region scores the value of the number, double the number, or triple the number, depending on which of the three parts the dart hits. If a person hits the bulls-eye with probability 0.01, hits a double with probability 0.10, hits a triple with probability 0.05, and misses the dart board with probability 0.02, what is the probability that 7 throws will result in no bulls-eyes, no triples, a double twice, and a complete miss once?

According to a genetics theory, a certain cross of guinea pigs will result in red, black, and white ospring in the ratio 8:4:4. Find the probability that among 8 ospring, 5 will be red, 2 black, and 1 white.

The probabilities are 0.4, 0.2, 0.3, and 0.1, respectively, that a delegate to a certain convention arrived by air, bus, automobile, or train. What is the probability that among 9 delegates randomly selected at this convention, 3 arrived by air, 3 arrived by bus, 1 arrived by automobile, and 2 arrived by train?

A safety engineer claims that only 40% of all workers wear safety helmets when they eat lunch at the workplace. Assuming that this claim is right, nd the probability that 4 of 6 workers randomly chosen will be wearing their helmets while having lunch at the workplace.

Suppose that for a very large shipment of integrated-circuit chips, the probability of failure for any one chip is 0.10. Assuming that the assumptions underlying the binomial distributions are met, nd the probability that at most 3 chips fail in a random sample of 20.

Assuming that 6 in 10 automobile accidents are due mainly to a speed violation, nd the probability that among 8 automobile accidents, 6 will be due mainly to a speed violation (a) by using the formula for the binomial distribution; (b) by using Table A.1.

If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights (a) exactly 18 will have a useful life of at least 800 hours; (b) at least 15 will have a useful life of at least 800 hours; (c) at least 2 will not have a useful life of at least 800 hours.

A manufacturer knows that on average 20% of the electric toasters produced require repairs within 1 year after they are sold. When 20 toasters are randomly selected, nd appropriate numbers x and y such that (a) the probability that at least x of them will require repairs is less than 0.5; (b) the probability that at least y of them will not require repairs is greater than 0.8.

A homeowner plants 6 bulbs selected at random from a box containing 5 tulip bulbs and 4 daffodil bulbs. What is the probability that he planted 2 daodil bulbs and 4 tulip bulbs?

To avoid detection at customs, a traveler places 6 narcotic tablets in a bottle containing 9 vitamin tablets that are similar in appearance. If the customs ocial selects 3 of the tablets at random for analysis, what is the probability that the traveler will be arrested for illegal possession of narcotics?

A random committee of size 3 is selected from 4 doctors and 2 nurses. Write a formula for the probability distribution of the random variable X representing the number of doctors on the committee. Find P(2 X 3).

From a lot of 10 missiles, 4 are selected at random and red. If the lot contains 3 defective missiles that will not re, what is the probability that (a) all 4 will re? (b) at most 2 will not re?

If 7 cards are dealt from an ordinary deck of 52 playing cards, what is the probability that (a) exactly 2 of them will be face cards? (b) at least 1 of them will be a queen?

What is the probability that a waitress will refuse to serve alcoholic beverages to only 2 minors if she randomly checks the IDs of 5 among 9 students, 4 of whom are minors?

A company is interested in evaluating its current inspection procedure for shipments of 50 identical items. The procedure is to take a sample of 5 and pass the shipment if no more than 2 are found to be defective. What proportion of shipments with 20% defectives will be accepted?

A manufacturing company uses an acceptance scheme on items from a production line before they are shipped. The plan is a two-stage one. Boxes of 25 items are readied for shipment, and a sample of 3 items is tested for defectives. If any defectives are found, the entire box is sent back for 100% screening. If no defectives are found, the box is shipped. (a) What is the probability that a box containing 3 defectives will be shipped? (b) What is the probability that a box containing only 1 defective will be sent back for screening?

Suppose that the manufacturing company of Exercise 5.36 decides to change its acceptance scheme. Under the new scheme, an inspector takes 1 item at random, inspects it, and then replaces it in the box; a second inspector does likewise. Finally, a third inspector goes through the same procedure. The box is not shipped if any of the three inspectors nd a defective. Answer the questions in Exercise 5.36 for this new plan.

Among 150 IRS employees in a large city, only 30 are women. If 10 of the employees are chosen at random to provide free tax assistance for the residents of this city, use the binomial approximation to the hypergeometric distribution to nd the probability that at least 3 women are selected.

An annexation suit against a county subdivision of 1200 residences is being considered by a neighboring city. If the occupants of half the residences object to being annexed, what is the probability that in a random sample of 10 at least 3 favor the annexation suit?

It is estimated that 4000 of the 10,000 voting residents of a town are against a new sales tax. If 15 eligible voters are selected at random and asked their opinion, what is the probability that at most 7 favor the new tax?

A nationwide survey of 17,000 college seniors by the University of Michigan revealed that almost 70% disapprove of daily pot smoking. If 18 of these seniors are selected at random and asked their opinion, what is the probability that more than 9 but fewer than 14 disapprove of smoking pot daily?

Find the probability of being dealt a bridge hand of 13 cards containing 5 spades, 2 hearts, 3 diamonds, and 3 clubs.

A foreign student club lists as its members 2 Canadians, 3 Japanese, 5 Italians, and 2 Germans. If a committee of 4 is selected at random, nd the probability that (a) all nationalities are represented; (b) all nationalities except Italian are represented.

An urn contains 3 green balls, 2 blue balls, and 4 red balls. In a random sample of 5 balls, find the probability that both blue balls and at least 1 red ball are selected.

Biologists doing studies in a particular environment often tag and release subjects in order to estimate the size of a population or the prevalence of certain features in the population. Ten animals of a certain population thought to be extinct (or near extinction) are caught, tagged, and released in a certain region. After a period of time, a random sample of 15 of this type of animal is selected in the region. What is the probability that 5 of those selected are tagged if there are 25 animals of this type in the region?

A large company has an inspection system for the batches of small compressors purchased from vendors. A batch typically contains 15 compressors. In the inspection system, a random sample of 5 is selected and all are tested. Suppose there are 2 faulty compressors in the batch of 15. (a) What is the probability that for a given sample there will be 1 faulty compressor? (b) What is the probability that inspection will discover both faulty compressors?

A government task force suspects that some manufacturing companies are in violation of federal pollution regulations with regard to dumping a certain type of product. Twenty firms are under suspicion but not all can be inspected. Suppose that 3 of the firms are in violation. (a) What is the probability that inspection of 5 firms will find no violations? (b) What is the probability that the plan above will find two violations?

Every hour, 10,000 cans of soda are filled by a machine, among which 300 underfilled cans are produced. Each hour, a sample of 30 cans is randomly selected and the number of ounces of soda per can is checked. Denote by X the number of cans selected that are underfilled. Find the probability that at least 1 underfilled can will be among those sampled.

The probability that a person living in a certain city owns a dog is estimated to be 0.3. Find the probability that the tenth person randomly interviewed in that city is the fth one to own a dog.

Find the probability that a person flipping a coin gets (a) the third head on the seventh flip; (b) the first head on the fourth flip.

Three people toss a fair coin and the odd one pays for coee. If the coins all turn up the same, they are tossed again. Find the probability that fewer than 4 tosses are needed.

A scientist inoculates mice, one at a time, with a disease germ until he nds 2 that have contracted the disease. If the probability of contracting the disease is 1/6, what is the probability that 8 mice are required?

An inventory study determines that, on average, demands for a particular item at a warehouse are made 5 times per day. What is the probability that on a given day this item is requested (a) more than 5 times? (b) not at all?

According to a study published by a group of University of Massachusetts sociologists, about twothirds of the 20 million persons in this country who take Valium are women. Assuming this gure to be a valid estimate, nd the probability that on a given day the fth prescription written by a doctor for Valium is (a) the rst prescribing Valium for a woman; (b) the third prescribing Valium for a woman.

The probability that a student pilot passes the written test for a private pilots license is 0.7. Find the probability that a given student will pass the test (a) on the third try; (b) before the fourth try.

On average, 3 trac accidents per month occur at a certain intersection. What is the probability that in any given month at this intersection (a) exactly 5 accidents will occur? (b) fewer than 3 accidents will occur? (c) at least 2 accidents will occur?

On average, a textbook author makes two wordprocessing errors per page on the rst draft of her textbook. What is the probability that on the next page she will make (a) 4 or more errors? (b) no errors?

A certain area of the eastern United States is, on average, hit by 6 hurricanes a year. Find the probability that in a given year that area will be hit by (a) fewer than 4 hurricanes; (b) anywhere from 6 to 8 hurricanes.

Suppose the probability that any given person will believe a tale about the transgressions of a famous actress is 0.8. What is the probability that (a) the sixth person to hear this tale is the fourth one to believe it? (b) the third person to hear this tale is the rst one to believe it?

The average number of eld mice per acre in a 5-acre wheat eld is estimated to be 12. Find the probability that fewer than 7 eld mice are found (a) on a given acre; (b) on 2 of the next 3 acres inspected.

Suppose that, on average, 1 person in 1000 makes a numerical error in preparing his or her income tax return. If 10,000 returns are selected at random and examined, nd the probability that 6, 7, or 8 of them contain an error.

The probability that a student at a local high school fails the screening test for scoliosis (curvature of the spine) is known to be 0.004. Of the next 1875 students at the school who are screened for scoliosis, nd the probability that (a) fewer than 5 fail the test; (b) 8, 9, or 10 fail the test.

Find the mean and variance of the random variable X in Exercise 5.58, representing the number of hurricanes per year to hit a certain area of the eastern United States.

Find the mean and variance of the random variable X in Exercise 5.61, representing the number of persons among 10,000 who make an error in preparing their income tax returns.

An automobile manufacturer is concerned about a fault in the braking mechanism of a particular model. The fault can, on rare occasions, cause a catastrophe at high speed. The distribution of the number of cars per year that will experience the catastrophe is a Poisson random variable with = 5. (a) What is the probability that at most 3 cars per year will experience a catastrophe? (b) What is the probability that more than 1 car per year will experience a catastrophe?

Changes in airport procedures require considerable planning. Arrival rates of aircraft are important factors that must be taken into account. Suppose small aircraft arrive at a certain airport, according to a Poisson process, at the rate of 6 per hour. Thus, the Poisson parameter for arrivals over a period of hours is =6 t. (a) What is the probability that exactly 4 small aircraft arrive during a 1-hour period? (b) What is the probability that at least 4 arrive during a 1-hour period? (c) If we dene a working day as 12 hours, what is the probability that at least 75 small aircraft arrive during a working day?

The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean \(\lambda = 7\). (a) Compute the probability that more than 10 customers will arrive in a 2-hour period. (b) What is the mean number of arrivals during a 2-hour period?

Consider Exercise 5.62. What is the mean number of students who fail the test?

The probability that a person will die when he or she contracts a virus infection is 0.001. Of the next 4000 people infected, what is the mean number who will die?

A company purchases large lots of a certain kind of electronic device. A method is used that rejects a lot if 2 or more defective units are found in a random sample of 100 units. (a) What is the mean number of defective units found in a sample of 100 units if the lot is 1% defective? (b) What is the variance?

For a certain type of copper wire, it is known that, on the average, 1.5 aws occur per millimeter. Assuming that the number of aws is a Poisson random variable, what is the probability that no aws occur in a certain portion of wire of length 5 millimeters? What is the mean number of aws in a portion of length 5 millimeters?

Potholes on a highway can be a serious problem, and are in constant need of repair. With a particular type of terrain and make of concrete, past experience suggests that there are, on the average, 2 potholes per mile after a certain amount of usage. It is assumed that the Poisson process applies to the random variable number of potholes. (a) What is the probability that no more than one pothole will appear in a section of 1 mile? (b) What is the probability that no more than 4 potholes will occur in a given section of 5 miles?

Hospital administrators in large cities anguish about trac in emergency rooms. At a particular hospital in a large city, the sta on hand cannot accom modate the patient trac if there are more than 10 emergency cases in a given hour. It is assumed that patient arrival follows a Poisson process, and historical data suggest that, on the average, 5 emergencies arrive per hour. (a) What is the probability that in a given hour the sta cannot accommodate the patient trac? (b) What is the probability that more than 20 emergencies arrive during a 3-hour shift?

It is known that 3% of people whose luggage is screened at an airport have questionable objects in their luggage. What is the probability that a string of 15 people pass through screening successfully before an individual is caught with a questionable object? What is the expected number of people to pass through before an individual is stopped?

Computer technology has produced an environment in which robots operate with the use of microprocessors. The probability that a robot fails during any 6-hour shift is 0.10. What is the probability that a robot will operate through at most 5 shifts before it fails?

The refusal rate for telephone polls is known to be approximately 20%. A newspaper report indicates that 50 people were interviewed before the rst refusal. (a) Comment on the validity of the report. Use a probability in your argument. (b) What is the expected number of people interviewed before a refusal?

During a manufacturing process, 15 units are randomly selected each day from the production line to check the percent defective. From historical information it is known that the probability of a defective unit is 0.05. Any time 2 or more defectives are found in the sample of 15, the process is stopped. This procedure is used to provide a signal in case the probability of a defective has increased. (a) What is the probability that on any given day the production process will be stopped? (Assume 5% defective.) (b) Suppose that the probability of a defective has increased to 0.07. What is the probability that on any given day the production process will not be stopped?

An automatic welding machine is being considered for use in a production process. It will be considered for purchase if it is successful on 99% of its welds. Otherwise, it will not be considered ecient. A test is to be conducted with a prototype that is to perform 100 welds. The machine will be accepted for manufacture if it misses no more than 3 welds. (a) What is the probability that a good machine will be rejected? (b) What is the probability that an inecient machine with 95% welding success will be accepted?

A car rental agency at a local airport has available 5 Fords, 7 Chevrolets, 4 Dodges, 3 Hondas, and 4 Toyotas. If the agency randomly selects 9 of these cars to chaueur delegates from the airport to the downtown convention center, nd the probability that 2 Fords, 3 Chevrolets, 1 Dodge, 1 Honda, and 2 Toyotas are used.

Service calls come to a maintenance center according to a Poisson process, and on average, 2.7 calls are received per minute. Find the probability that (a) no more than 4 calls come in any minute; (b) fewer than 2 calls come in any minute; (c) more than 10 calls come in a 5-minute period.

An electronics firm claims that the proportion of defective units from a certain process is 5%. A buyer has a standard procedure of inspecting 15 units selected randomly from a large lot. On a particular occasion, the buyer found 5 items defective. (a) What is the probability of this occurrence, given that the claim of 5% defective is correct? (b) What would be your reaction if you were the buyer?

An electronic switching device occasionally malfunctions, but the device is considered satisfactory if it makes, on average, no more than 0.20 error per hour. A particular 5-hour period is chosen for testing the device. If no more than 1 error occurs during the time period, the device will be considered satisfactory. (a) What is the probability that a satisfactory device will be considered unsatisfactory on the basis of the test? Assume a Poisson process. (b) What is the probability that a device will be accepted as satisfactory when, in fact, the mean number of errors is 0.25? Again, assume a Poisson process.

A company generally purchases large lots of a certain kind of electronic device. A method is used that rejects a lot if 2 or more defective units are found in a random sample of 100 units. (a) What is the probability of rejecting a lot that is 1% defective? (b) What is the probability of accepting a lot that is 5% defective?

A local drugstore owner knows that, on average, 100 people enter his store each hour. (a) Find the probability that in a given 3-minute period nobody enters the store. (b) Find the probability that in a given 3-minute period more than 5 people enter the store.

(a) Suppose that you throw 4 dice. Find the probability that you get at least one 1. (b) Suppose that you throw 2 dice 24 times. Find the probability that you get at least one (1, 1), that is, snake-eyes.

Suppose that out of 500 lottery tickets sold, 200 pay o at least the cost of the ticket. Now suppose that you buy 5 tickets. Find the probability that you will win back at least the cost of 3 tickets.

Imperfections in computer circuit boards and computer chips lend themselves to statistical treatment. For a particular type of board, the probability of a diode failure is 0.03 and the board contains 200 diodes. (a) What is the mean number of failures among the diodes? (b) What is the variance? (c) The board will work if there are no defective diodes. What is the probability that a board will work?

The potential buyer of a particular engine requires (among other things) that the engine start successfully 10 consecutive times. Suppose the probability of a successful start is 0.990. Let us assume that the outcomes of attempted starts are independent. (a) What is the probability that the engine is accepted after only 10 starts? (b) What is the probability that 12 attempted starts are made during the acceptance process?

The acceptance scheme for purchasing lots containing a large number of batteries is to test no more than 75 randomly selected batteries and to reject a lot if a single battery fails. Suppose the probability of a failure is 0.001. (a) What is the probability that a lot is accepted? (b) What is the probability that a lot is rejected on the 20th test? (c) What is the probability that it is rejected in 10 or fewer trials?

An oil drilling company ventures into various locations, and its success or failure is independent from one location to another. Suppose the probability of a success at any specic location is 0.25. (a) What is the probability that the driller drills at 10 locations and has 1 success? (b) The driller will go bankrupt if it drills 10 times before the rst success occurs. What are the drillers prospects for bankruptcy?

Consider the information in Review Exercise 5.90. The drilling company feels that it will hit it big if the second success occurs on or before the sixth attempt. What is the probability that the driller will hit it big?

A couple decides to continue to have children until they have two males. Assuming that P(male) = 0.5, what is the probability that their second male is their fourth child?

It is known by researchers that 1 in 100 people carries a gene that leads to the inheritance of a certain chronic disease. In a random sample of 1000 individuals, what is the probability that fewer than 7 individuals carry the gene? Use a Poisson approximation. Again, using the approximation, what is the approximate mean number of people out of 1000 carrying the gene?

A production process produces electronic component parts. It is presumed that the probability of a defective part is 0.01. During a test of this presumption, 500 parts are sampled randomly and 15 defectives are observed. (a) What is your response to the presumption that the process is 1% defective? Be sure that a computed probability accompanies your comment. (b) Under the presumption of a 1% defective process, what is the probability that only 3 parts will be found defective? (c) Do parts (a) and (b) again using the Poisson approximation.

A production process outputs items in lots of 50. Sampling plans exist in which lots are pulled aside periodically and exposed to a certain type of inspection. It is usually assumed that the proportion defective is very small. It is important to the company that lots containing defectives be a rare event. The current inspection plan is to periodically sample randomly 10 out of the 50 items in a lot and, if none are defective, to perform no intervention. (a) Suppose in a lot chosen at random, 2 out of 50 are defective. What is the probability that at least 1 in the sample of 10 from the lot is defective? (b) From your answer to part (a), comment on the quality of this sampling plan. (c) What is the mean number of defects found out of 10 items sampled?

Consider the situation of Review Exercise 5.95. It has been determined that the sampling plan should be extensive enough that there is a high probability, say 0.9, that if as many as 2 defectives exist in the lot of 50 being sampled, at least 1 will be found in the sampling. With these restrictions, how many of the 50 items should be sampled?

National security requires that defense technology be able to detect incoming projectiles or missiles. To make the defense system successful, multiple radar screens are required. Suppose that three independent screens are to be operated and the probability that any one screen will detect an incoming missile is 0.8. Obviously, if no screens detect an incoming projectile, the system is unworthy and must be improved. (a) What is the probability that an incoming missile will not be detected by any of the three screens? (b) What is the probability that the missile will be detected by only one screen? (c) What is the probability that it will be detected by at least two out of three screens?

Suppose it is important that the overall missile defense system be as near perfect as possible. (a) Assuming the quality of the screens is as indicated in Review Exercise 5.97, how many are needed to ensure that the probability that a missile gets through undetected is 0.0001? (b) Suppose it is decided to stay with only 3 screens and attempt to improve the screen detection ability. What must the individual screen eectiveness (i.e., probability of detection) be in order to achieve the eectiveness required in part (a)?

Go back to Review Exercise 5.95(a). Recompute the probability using the binomial distribution. Comment.

There are two vacancies in a certain university statistics department. Five individuals apply. Two have expertise in linear models, and one has expertise in applied probability. The search committee is instructed to choose the two applicants randomly. (a) What is the probability that the two chosen are those with expertise in linear models? (b) What is the probability that of the two chosen, one has expertise in linear models and one has expertise in applied probability?

The manufacturer of a tricycle for children has received complaints about defective brakes in the product. According to the design of the product and considerable preliminary testing, it had been determined that the probability of the kind of defect in the complaint was 1 in 10,000 (i.e., 0.0001). After a thorough investigation of the complaints, it was determined that during a certain period of time, 200 products were randomly chosen from production and 5 had defective brakes. (a) Comment on the “1 in 10,000” claim by the manufacturer. Use a probabilistic argument. Use the binomial distribution for your calculations. (b) Repeat part (a) using the Poisson approximation.

Group Project: Divide the class into two groups of approximately equal size. The students in group 1 will each toss a coin 10 times (n1) and count the number of heads obtained. The students in group 2 will each toss a coin 40 times (n2) and again count the number of heads. The students in each group should individually compute the proportion of heads observed, which is an estimate of p, the probability of observing a head. Thus, there will be a set of values of p1 (from group 1) and a set of values p2 (from group 2). All of the values of p1 and p2 are estimates of 0.5, which is the true value of the probability of observing a head for a fair coin. (a) Which set of values is consistently closer to 0.5, the values of p1 or p2? Consider the proof of Theorem 5.1 on page 147 with regard to the estimates of the parameter p =0 .5. The values of p1 were obtained with n = n1 = 10, and the values of p2 were obtained with n = n2 = 40. Using the notation of the proof, the estimates are given by p1 = x1 n1 = I1 ++ In1 n1 , where I1,...,In1 are 0s and 1s and n1 = 10, and p2 = x2 n2 = I1 ++ In2 n2 , where I1,...,In2, again, are 0s and 1s and n2 = 40. (b) Referring again to Theorem 5.1, show that E(p1)=E(p2)=p =0 .5. (c) Show that 2 p1 = 2 X1 n1 is 4 times the value of 2 p2 = 2 X2 n2 . Then explain further why the values of p2 from group 2 are more consistently closer to the true value, p =0 .5, than the values of p1 from group 1. You will continue to learn more and more about parameter estimation beginning in Chapter 9. At that point emphasis will put on the importance of the mean and variance of an estimator of a parameter.