Biologists doing studies in a particular environment often tag and release subjects in order to estimate 158 Chapter 5 Some Discrete Probability Distributions 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?
> 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
> 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 probabil
> 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.
> 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 am
> Suppose that out of 500 lottery tickets sold, 200 pay off 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.
> (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.”
> 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
> 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% defecti
> 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 dur
> 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) W
> 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
> 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
> 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 chauffeur delegates from the airport to the downtown convention center, find the probability
> 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 efficient. A test is to be conducted with a prototype that
> 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 f
> The refusal rate for telephone polls is known to be approximately 20%. A newspaper report indicates that 50 people were interviewed before the first refusal. (a) Comment on the validity of the report. Use a probability in your argument. (b) What is the e
> 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 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 ob
> Hospital administrators in large cities anguish about traffic in emergency rooms. At a particular hospital in a large city, the staff on hand cannot accommodate the patient traffic if there are more than 10 emergency cases in a given hour. It is assumed
> 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 i
> For a certain type of copper wire, it is known that, on the average, 1.5 flaws occur per millimeter. Assuming that the number of flaws is a Poisson random variable, what is the probability that no flaws occur in a certain portion of wire of length 5 mill
> 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 uni
> One prominent physician claims that 70% of those with lung cancer are chain smokers. If his assertion is correct, (a) find the probability that of 10 such patients recently admitted to a hospital, fewer than half are chain smokers; (b) find the probabili
> 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?
> Consider Exercise 5.62. What is the mean number of students who fail the test? Exercise 5.62: 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 st
> The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean λ = 7. (a) Compute the probability that more than 10 customers will arrive in a 2-hour period. (b) What is the mean n
> 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. T
> 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 catastroph
> 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.
> 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.
> 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, find the probability that (a) fewer than 5
> 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, find the probability that 6, 7, or 8 of them contain an error.
> The average number of field mice per acre in a 5-acre wheat field is estimated to be 12. Find the probability that fewer than 7 field mice are found (a) on a given acre; (b) on 2 of the next 3 acres inspected.
> 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 t
> 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
> 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.
> On average, a textbook author makes two word processing errors per page on the first draft of her textbook. What is the probability that on the next page she will make (a) 4 or more errors? (b) no errors?
> On average, 3 traffic 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
> The probability that a student pilot passes the written test for a private pilot’s 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.
> According to a study published by a group of University of Massachusetts sociologists, about two thirds of the 20 million persons in this country who take Valium are women. Assuming this figure to be a valid estimate, find the probability that on a given
> 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?
> A scientist inoculates mice, one at a time, with a disease germ until he finds 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?
> Three people toss a fair coin and the odd one pays for coffee. If the coins all turn up the same, they are tossed again. Find the probability that fewer than 4 tosses are needed.
> 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.
> 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
> 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 fifth one to own a dog.
> 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
> 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 fi
> 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 com
> 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.
> Consider a ferry that can carry both buses and cars across a waterway. Each trip costs the owner approximately $10. The fee for cars is $3 and the fee for buses is $8. Let X and Y denote the number of buses and cars, respectively, carried on a given trip
> A convenience store has two separate locations where customers can be checked out as they leave. These locations each have two cash registers and two employees who check out customers. Let X be the number of cash registers being used at a particular time
> A delivery truck travels from point A to point B and back using the same route each day. There are four traffic lights on the route. Let X1 denote the number of red lights the truck encounters going from A to B and X2 denote the number encountered on the
> It is known through data collection and considerable research that the amount of time in seconds that a certain employee of a company is late for work is a random variable X with density function In other words, he not only is slightly late at times, b
> In business, it is important to plan and carry out research in order to anticipate what will occur at the end of the year. Research suggests that the profit (loss) spectrum for a certain company, with corresponding probabilities, is as follows: (a) Wha
> In a support system in the U.S. space program, a single crucial component works only 85% of the time. In order to enhance the reliability of the system, it is decided that 3 components will be installed in parallel such that the system fails only if they
> A company’s marketing and accounting departments have determined that if the company markets its newly developed product, the contribution of the product to the firm’s profit during the next 6 months will be described
> Consider Exercise 4.10 on page 117. Can it be said that the ratings given by the two experts are independent? Explain why or why not. Exercise 4.10: Two tire-quality experts examine stacks of tires and assign a quality rating to each tire on a 3-point s
> A dealer’s profit, in units of $5000, on a new automobile is a random variable X having density function (a) Find the variance of the dealer’s profit. (b) Demonstrate that Chebyshev’s theorem holds
> Consider random variables X and Y of Exercise 4.63 on page 138. Compute ρXY. Exercise 4.63: Repeat Exercise 4.62 if X and Y are not independent and σXY = 1. Exercise 4.62: If X and Y are independent random variables with variances σ2X = 5 and σ2Y = 3, f
> 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, find the probability that (a) all nationalities are represented; (b) all nationalities except Italian are represent
> A private pilot wishes to insure his airplane for $200,000. The insurance company estimates that a total loss will occur with probability 0.002, a 50% loss with probability 0.01, and a 25% loss with probability 0.1. Ignoring all other partial losses, wha
> Consider the joint density function Compute the correlation coefficient ρXY. 16у , x > 2, 0 < y < 1, (0, f (x, y) = elsewhere.
> Consider the density function of Review Exercise 4.85. Demonstrate that Chebyshev’s theorem holds for k = 2 and k = 3. Review Exercise 4.85: Suppose it is known that the life X of a particular compressor, in hours, has the density func
> Show that Cov(aX, bY ) = ab Cov(X, Y ).
> Referring to the random variables whose joint density function is given in Exercise 3.40 on page 105, (a) find μX and μY ; (b) find E[(X + Y )/2].
> Suppose it is known that the life X of a particular compressor, in hours, has the density function (a) Find the mean life of the compressor. (b) Find E(X2). (c) Find the variance and standard deviation of the random variable X. 1/900 e 900 x > 0, f
> Referring to the random variables whose joint probability density function is given in Exercise 3.41 on page 105, find the expected weight for the sum of the creams and toffees if one purchased a box of these chocolates.
> Referring to the random variables whose joint density function is given in Exercise 3.41 on page 105, find the covariance between the weight of the creams and the weight of the toffees in these boxes of chocolates.
> Assume the length X, in minutes, of a particular type of telephone conversation is a random variable with probability density function (a) Determine the mean length E(X) of this type of telephone conversation. (b) Find the variance and standard deviati
> Referring to the random variables whose joint probability density function is given in Exercise 3.47 on page 105, find the average amount of kerosene left in the tank at the end of the day.
> Find the probability of being dealt a bridge hand of 13 cards containing 5 spades, 2 hearts, 3 diamonds, and 3 clubs.
> Find the covariance of random variables X and Y having the joint probability density function Sx + y, 0< x < 1, 0 < y < 1, [0, f(x, y) elsewhere.
> Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the probabilities are 0.22, 0.36, 0.28, and 0.14, respectively, that she will be able to sell it for a profit of $250, sell it for a profit of $150, break even,
> Prove Chebyshev’s theorem.
> Compute P(μ − 2σ and compare with the result given in Chebyshev’s theorem. J бх (1 — а), 0 <х <1, 0, f (x) = elsewhere,
> A random variable X has a mean μ = 10 and a variance σ2 = 4. Using Chebyshev’s theorem, find (a) P(|X − 10| ≥ 3); (b) P(|X − 10| < 3); (c) P(5
> Seventy new jobs are opening up at an automobile manufacturing plant, and 1000 applicants show up for the 70 positions. To select the best 70 from among the applicants, the company gives a test that covers mechanical skill, manual dexterity, and mathemat
> An electrical firm manufactures a 100-watt light bulb, which, according to specifications written on the package, has a mean life of 900 hours with a standard deviation of 50 hours. At most, what percentage of the bulbs fail to last even 700 hours? Assum
> Consider again the situation of Exercise 4.72. It is required to find Var(eY ). Use Theorems 4.2 and 4.3 and define Z = eY . Thus, use the conditions of Exercise 4.73 to find Var(Z) = E(Z2 ) − [E(Z)]2 .
> For the situation in Exercise 4.72, compute E(eY ) using Theorem 4.1, that is, by using Then compute E(eY ) not by using f(y), but rather by using the second-order adjustment to the first-order approximation of E(eY ). Comment. E(e*) = | e" f(y) dy
> A manufacturing company has developed a machine for cleaning carpet that is fuel-efficient because it delivers carpet cleaner so rapidly. Of interest is a random variable Y, the amount in gallons per minute delivered. It is known that the density functio
> 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 fewe
> The length of time Y, in minutes, required to generate a human reflex to tear gas has the density function (a) What is the mean time to reflex? (b) Find E(Y2) and Var(Y ). Sie-/4, 0<y < , f(y) [ 0, elsewhere.
> Consider Review Exercise 3.64 on page 107. There are two service lines. The random variables X and Y are the proportions of time that line 1 and line 2 are in use, respectively. The joint probability density function for (X, Y ) is given by (a) Determi
> By investing in a particular stock, a person can make a profit in one year of $4000 with probability 0.3 or take a loss of $1000 with probability 0.7. What is this person’s expected gain?
> Consider Review Exercise 3.77 on page 108. The random variables X and Y represent the number of vehicles that arrive at two separate street corners during a certain 2-minute period in the day. The joint distribution is (a) Give E(X), E(Y ), Var(X), and
> The power P in watts which is dissipated in an electric circuit with resistance R is known to be given by P = I2R, where I is current in amperes and R is a constant fixed at 50 ohms. However, I is a random variable with μI = 15 amperes and σ2I = 0.03 amp
> If the joint density function of X and Y is given by find the expected value of S{ 0 < x < 1, 1< y < 2, (x + 2y), f (x, y) = 10, elsewhere, g() = + X?Y. X, Y
> Let X represent the number that occurs when a green die is tossed and Y the number that occurs when a red die is tossed. Find the variance of the random variable (a) 2X − Y; (b) X + 3Y − 5.
> Let X represent the number that occurs when a red die is tossed and Y the number that occurs when a green die is tossed. Find (a) E(X + Y ); (b) E(X − Y ); (c) E(XY ).
> Suppose that X and Y are independent random variables with probability densities and and Find the expected value of Z = XY . x > 2, [0, g(x) elsewhere, 2y, 0<y < 1, h(y) 0, elsewhere.
> Repeat Exercise 4.62 if X and Y are not independent and σXY = 1. Exercise 4.62: If X and Y are independent random variables with variances σ2X = 5 and σ2Y = 3, find the variance of the random variable Z = −2X + 4Y − 3.
> 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?
> 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)
> If X and Y are independent random variables with variances σ2X = 5 and σ2Y = 3, find the variance of the random variable Z = −2X + 4Y − 3.
> Use Theorem 4.7 to evaluate E(2XY2 − X2Y ) for the joint probability distribution shown in Table 3.1 on page 96.
> Suppose that X and Y are independent random variables having the joint probability distribution Find (a) E(2X − 3Y ); (b) E(XY ). f(x, y) 4 0.10 0.15 3 0.20 1 0.30 0.10 0.15
> An attendant at a car wash is paid according to the number of cars that pass through. Suppose the probabilities are 1/12, 1/12, 1/4, 1/4, 1/6, and 1/6, respectively, that the attendant receives $7, $9, $11, $13, $15, or $17 between 4:00 P.M. and 5:00 P.M
> If a random variable X is defined such that E[(X − 1)2] = 10 and E[(X − 2)2 ]=6, find μ and σ2.
