Question: From past experience, a stockbroker believes that

> An overseas shipment of 5 foreign automobiles contains 2 that have slight paint blemishes. If an agency receives 3 of these automobiles at random, list the elements of the sample space S, using the letters B and N for blemished and non-blemished, respect

> For the density function of Exercise 3.17, find F(x). Use it to evaluate P(2 < X < 2.5). Exercise 3.17: A continuous random variable X that can assume values between x = 1 and x = 3 has a density function given by f(x) = 1/2.

> 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, find the probability that at most 3 chips fail in a rand

> A continuous random variable X that can assume values between x = 2 and x = 5 has a density function given by f(x) = 2(1 + x)/27. Find (a) P(X

> A continuous random variable X that can assume values between x = 1 and x = 3 has a density function given by f(x) = 1/2. (a) Show that the area under the curve is equal to 1. (b) Find P(2 < X < 2.5). (c) Find P(X ≤ 1.6).

> Construct a graph of the cumulative distribution function of Exercise 3.15.

> Find the cumulative distribution function of the random variable X representing the number of defectives in Exercise 3.11. Then using F(x), find (a) P(X = 1); (b) P(0 < X ≤ 2).

> The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution function Find the probability of waiting less than 12 minutes between successive speeders (a) using the cumulat

> The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given by Construct the cumulative distribution function of X. 1 3 4 f(x) 0.41 0.37 0.16 0.05 0.01

> An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function of T, the number of years to maturity for a randomly selected bond, is find (a) P(T = 5); (b) P(T &gt; 3)

> A shipment of 7 television sets contains 2 defective sets. A hotel makes a random purchase of 3 of the sets. If x is the number of defective sets purchased by the hotel, find the probability distribution of X. Express the results graphically as a probabi

> A paint-store chain produces and sells latex and semigloss paint. Based on long-range sales, the probability that a customer will purchase latex paint is 0.75. Of those that purchase latex paint, 60% also purchase rollers. But only 30% of semigloss paint

> A regional telephone company operates three identical relay stations at different locations. During a one-year period, the number of malfunctions reported by each station and the causes are shown below. Suppose that a malfunction was reported and it wa

> 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, find the probability that 4 of 6 workers randomly chosen will be wearing their helmets while having lunch a

> Find a formula for the probability distribution of the random variable X representing the outcome when a single die is rolled once.

> Classify the following random variables as discrete or continuous: X: the number of automobile accidents per year in Virginia. Y : the length of time to play 18 holes of golf. M: the amount of milk produced yearly by a particular cow. N: the number of eg

> For the sample space of Exercise 2.5, (a) list the elements corresponding to the event A that a number less than 3 occurs on the die; (b) list the elements corresponding to the event B that two tails occur; (c) list the elements corresponding to the even

> The probability that an automobile being filled with gasoline also needs an oil change is 0.25; the probability that it needs a new oil filter is 0.40; and the probability that both the oil and the filter need changing is 0.14. (a) If the oil has to be c

> For the sample space of Exercise 2.4, (a) list the elements corresponding to the event A that the sum is greater than 8; (b) list the elements corresponding to the event B that a 2 occurs on either die; (c) list the elements corresponding to the event C

> In USA Today (Sept. 5, 1996), the results of a survey involving the use of sleepwear while traveling were listed as follows: (a) What is the probability that a traveler is a female who sleeps in the nude? (b) What is the probability that a traveler is

> A manufacturer of a flu vaccine is concerned about the quality of its flu serum. Batches of serum are processed by three different departments having rejection rates of 0.10, 0.08, and 0.12, respectively. The inspections by the three departments are sequ

> In the senior year of a high school graduating class of 100 students, 42 studied mathematics, 68 studied psychology, 54 studied history, 22 studied both mathematics and history, 25 studied both mathematics and psychology, 7 studied history but neither ma

> In an experiment to study the relationship of hypertension and smoking habits, the following data are collected for 180 individuals: where H and NH in the table stand for Hypertension and Nonhypertension, respectively. If one of these individuals is se

> A random sample of 200 adults are classified below by sex and their level of education attained. If a person is picked at random from this group, find the probability that (a) the person is a male, given that the person has a secondary education; (b) t

> 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 arrive

> A class in advanced physics is composed of 10 juniors, 30 seniors, and 10 graduate students. The final grades show that 3 of the juniors, 10 of the seniors, and 5 of the graduate students received an A for the course. If a student is chosen at random fro

> If R is the event that a convict committed armed robbery and D is the event that the convict pushed dope, state in words what probabilities are expressed by (a) P(R|D); (b) P(D’|R); (c) P(R’|D’).

> Prove that P(A’ ∩ B’) = 1+P(A ∩ B) − P(A) − P(B).

> As the situation of Exercise 2.69 might suggest, statistical procedures are often used for control of quality (i.e., industrial quality control). At times, the weight of a product is an important variable to control. Specifications are given for the weig

> Consider the situation of Exercise 2.69. Suppose 50,000 boxes of detergent are produced per week and suppose also that those underfilled are “sent back,” with customers requesting reimbursement of the purchase price. Suppose also that the cost of product

> Four students are selected at random from a chemistry class and classified as male or female. List the elements of the sample space S1, using the letter M for male and F for female. Define a second sample space S2 where the elements represent the number

> It is common in many industrial areas to use a filling machine to fill boxes full of product. This occurs in the food industry as well as other areas in which the product is used in the home, for example, detergent. These machines are not perfect, and in

> Interest centers around the nature of an oven purchased at a particular department store. It can be either a gas or an electric oven. Consider the decisions made by six distinct customers. (a) Suppose that the probability is 0.40 that at most two of thes

> Consider the situation of Example 2.32 on page 58. (a) What is the probability that no more than 4 cars will be serviced by the mechanic? (b) What is the probability that he will service fewer than 8 cars? (c) What is the probability that he will service

> Factory workers are constantly encouraged to practice zero tolerance when it comes to accidents in factories. Accidents can occur because the working environment or conditions themselves are unsafe. On the other hand, accidents can occur due to carelessn

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

> Consider the situation of Exercise 2.64. Let A be the event that the component fails a particular test and B be the event that the component displays strain but does not actually fail. Event A occurs with probability 0.20, and event B occurs with probabi

> Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is

> According to Consumer Digest (July/August 1996), the probable location of personal computers (PC) in the home is as follows: Adult bedroom: ………………………………. 0.03 Child bedroom: ……………………………….. 0.15 Other bedroom: ………………………………. 0.14 Office or den: …………………………

> Dom’s Pizza Company uses taste testing and statistical analysis of the data prior to marketing any new product. Consider a study involving three types of crusts (thin, thin with garlic and oregano, and thin with bits of cheese). Dom’s is also studying th

> In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the probability that (a) the student took mathematics or hist

> If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary, what is the probability that (a) the dictionary is selected? (b) 2 novels and 1 book of poems are selected?

> Two jurors are selected from 4 alternates to serve at a murder trial. Using the notation A1A3, for example, to denote the simple event that alternates 1 and 3 are selected, list the 6 elements of the sample space S.

> In a poker hand consisting of 5 cards, find the probability of holding (a) 3 aces; (b) 4 hearts and 1 club.

> A pair of fair dice is tossed. Find the probability of getting (a) a total of 8; (b) at most a total of 5.

> If a letter is chosen at random from the English alphabet, find the probability that the letter (a) is a vowel exclusive of y; (b) is listed somewhere ahead of the letter j; (c) is listed somewhere after the letter g.

> The surface of a circular dart board has a small center circle called the bull’s-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 sp

> An automobile manufacturer is concerned about a possible recall of its best-selling four-door sedan. If there were a recall, there is a probability of 0.25 of a defect in the brake system, 0.18 of a defect in the transmission, 0.17 of a defect in the fue

> If each coded item in a catalog begins with 3 distinct letters followed by 4 distinct nonzero digits, find the probability of randomly selecting one of these coded items with the first letter a vowel and the last digit even.

> The probability that an American industry will locate in Shanghai, China, is 0.7, the probability that it will locate in Beijing, China, is 0.4, and the probability that it will locate in either Shanghai or Beijing or both is 0.8. What is the probability

> Suppose that in a senior college class of 500 students it is found that 210 smoke, 258 drink alcoholic beverages, 216 eat between meals, 122 smoke and drink alcoholic beverages, 83 eat between meals and drink alcoholic beverages, 97 smoke and eat between

> A box contains 500 envelopes, of which 75 contain $100 in cash, 150 contain $25, and 275 contain $10. An envelope may be purchased for $25. What is the sample space for the different amounts of money? Assign probabilities to the sample points and then fi

> Assuming that all elements of S in Exercise 2.8 on page 42 are equally likely to occur, find (a) the probability of event A; (b) the probability of event C; (c) the probability of event A ∩ C.

> An experiment consists of tossing a die and then flipping a coin once if the number on the die is even. If the number on the die is odd, the coin is flipped twice. Using the notation 4H, for example, to denote the outcome that the die comes up 4 and then

> Find the errors in each of the following statements: (a) The probabilities that an automobile salesperson will sell 0, 1, 2, or 3 cars on any given day in February are, respectively, 0.19, 0.38, 0.29, and 0.15. (b) The probability that it will rain tomor

> How many ways are there that no two students will have the same birth date in a class of size 60?

> 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

> 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 speake

> How many ways are there to select 3 candidates from 8 equally qualified recent graduates for openings in an accounting firm?

> In how many ways can 3 oaks, 4 pines, and 2 maples be arranged along a property line if one does not distinguish among trees of the same kind?

> How many distinct permutations can be made from the letters of the word INFINITY?

> In how many ways can a caravan of 8 covered wagons from Arizona be arranged in a circle?

> In how many ways can 5 different trees be planted in a circle?

> Three lottery tickets for first, second, and third prizes are drawn from a group of 40 tickets. Find the number of sample points in S for awarding the 3 prizes if each contestant holds only 1 ticket.

> Find the number of ways that 6 teachers can be assigned to 4 sections of an introductory psychology course if no teacher is assigned to more than one section.

> In how many ways can 5 starting positions on a basketball team be filled with 8 men who can play any of the positions?

> An experiment involves tossing a pair of dice, one green and one red, and recording the numbers that come up. If x equals the outcome on the green die and y the outcome on the red die, describe the sample space S (a) by listing the elements (x, y); (b) b

> In a regional spelling bee, the 8 finalists consist of 3 boys and 5 girls. Find the number of sample points in the sample space S for the number of possible orders at the conclusion of the contest for (a) all 8 finalists; (b) the first 3 positions.

> 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 X1 be the numb

> Four married couples have bought 8 seats in the same row for a concert. In how many different ways can they be seated (a) with no restrictions? (b) if each couple is to sit together? (c) if all the men sit together to the right of all the women?

> In how many ways can 4 boys and 5 girls sit in a row if the boys and girls must alternate?

> (a) How many three-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, and 6 if each digit can be used only once? (b) How many of these are odd numbers? (c) How many are greater than 330?

> A contractor wishes to build 9 houses, each different in design. In how many ways can he place these houses on a street if 6 lots are on one side of the street and 3 lots are on the opposite side?

> (a) How many distinct permutations can be made from the letters of the word COLUMNS? (b) How many of these permutations start with the letter M?

> If a multiple-choice test consists of 5 questions, each with 4 possible answers of which only 1 is correct, (a) in how many different ways can a student check off one answer to each question? (b) in how many ways can a student check off one answer to ea

> (a) In how many ways can 6 people be lined up to get on a bus? (b) If 3 specific persons, among 6, insist on following each other, how many ways are possible? (c) If 2 specific persons, among 6, refuse to follow each other, how many ways are possible?

> A witness to a hit-and-run accident told the police that the license number contained the letters RLH followed by 3 digits, the first of which was a 5. If the witness cannot recall the last 2 digits, but is certain that all 3 digits are different, find t

> In how many different ways can a true-false test consisting of 9 questions be answered?

> Which of the following events are equal? (a) A = {1, 3}; (b) B = {x | x is a number on a die}; (c) C = {x | x2 − 4x+3 = 0}; (d) D = {x | x is the number of heads when six coins are tossed}.

> (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? Data from Exercise 5.9: In testing a certain kind of truck tire over

> In a fuel economy study, each of 3 race cars is tested using 5 different brands of gasoline at 7 test sites located in different regions of the country. If 2 drivers are used in the study, and test runs are made once under each distinct set of conditions

> A drug for the relief of asthma can be purchased from 5 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. How many different ways can a doctor prescribe the drug for a patient suffering from asth

> A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different plans are available to this buyer?

> A California study concluded that following 7 simple health rules can extend a man’s life by 11 years on the average and a woman’s life by 7 years. These 7 rules are as follows: no smoking, get regular exercise, use alcohol only in moderation, get 7 to 8

> A certain brand of shoes comes in 5 different styles, with each style available in 4 distinct colors. If the store wishes to display pairs of these shoes showing all of its various styles and colors, how many different pairs will the store have on displa

> Students at a private liberal arts college are classified as being freshmen, sophomores, juniors, or seniors, and also according to whether they are male or female. Find the total number of possible classifications for the students of that college.

> If an experiment consists of throwing a die and then drawing a letter at random from the English alphabet, how many points are there in the sample space?

> In a medical study, patients are classified in 8 ways according to whether they have blood type AB+, AB− , A+, A−, B+, B− , O+, or O−, and also according to whether their blood pressure is low, normal, or high. Find the number of ways in which a patient

> Registrants at a large convention are offered 6 sightseeing tours on each of 3 days. In how many ways can a person arrange to go on a sightseeing tour planned by this convention?

> Referring to Exercise 2.19 and the Venn diagram of Figure 2.5, list the numbers of the regions that represent the following events: (a) The family will experience no mechanical problems and will not receive a ticket for a traffic violation but will arriv

> 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. Exercise 5.13: A national study that examined at

> According to the journal Chemical Engineering, an important property of a fiber is its water absorbency. A random sample of 20 pieces of cotton fiber was taken and the absorbency on each piece was measured. The following are the absorbency values: (a)

> Suppose that a family is leaving on a summer vacation in their camper and that M is the event that they will experience mechanical problems, T is the event that they will receive a ticket for committing a traffic violation, and V is the event that they w

> Which of the following pairs of events are mutually exclusive? (a) A golfer scoring the lowest 18-hole round in a 72- hole tournament and losing the tournament. (b) A poker player getting a flush (all cards in the same suit) and 3 of a kind on the same 5

> Let A, B, and C be events relative to the sample space S. Using Venn diagrams, shade the areas representing the following events: (a) (A ∩ B)’; (b) (A ∪ B)’; (c) (A ∩ C) ∪ B.

> If S = {x | 0 < x < 12}, M = {x | 1 < x < 9}, and N = {x | 0 < x < 5}, find (a) M ∪ N; (b) M ∩ N; (c) M’ ∩ N’.

> Consider the sample space S = {copper, sodium, nitrogen, potassium, uranium, oxygen, zinc} and the events A = {copper, sodium, zinc}, B = {sodium, nitrogen, potassium}, C = {oxygen}. List the elements of the sets corresponding to the following events: (a

> If S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {0, 2, 4, 6, 8}, B = {1, 3, 5, 7, 9}, C = {2, 3, 4, 5}, and D = {1, 6, 7}, list the elements of the sets corresponding to the following events: (a) A ∪ C; (b) A ∩ B; (c) C’; (d) (C’ ∩ D) ∪ B; (e) (S ∩ C)’; (f

> Construct a Venn diagram to illustrate the possible intersections and unions for the following events relative to the sample space consisting of all automobiles made in the United States. F: Four door, S: Sun roof, P: Power steering.

> Exercise and diet are being studied as possible substitutes for medication to lower blood pressure. Three groups of subjects will be used to study the effect of exercise. Group 1 is sedentary, while group 2 walks and group 3 swims for 1 hour a day. Half