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All Textbook Solutions for Mathematical Statistics with Applications

Prove that the sum of the deviations of a set of measurements about their mean is equal to zero; that is,The mean duration of television commercials is 75 seconds with standard deviation 20 seconds. Assume that the durations are approximately normally distributed to answer the following. a What percentage of commercials last longer than 95 seconds? b What percentage of the commercials last between 35 and 115 seconds? c Would you expect commercial to last longer than 2 minutes? Why or why not?Aqua running has been suggested as a method of cardiovascular conditioning for injured athletes and others who desire a low-impact aerobics program. In a study to investigate the relationship between exercise cadence and heart rate,1 the heart rates of 20 healthy volunteers were measured at a cadence of 48 cycles per minute (a cycle consisted of two steps). The data are as follows: a Use the range of the measurements to obtain an estimate of the standard deviation. b Construct a frequency histogram for the data. Use the histogram to retain a visual approximation to y and s. c Calculate y and s. Compare these results with the calculation checks provided by parts (a) and (b). d Construct the intervals yks,, k = 1, 2, and 3, and count the number of measurements falling in each interval. Compare the fractions falling in the intervals with the fractions that you would expect according to the empirical rule.The following data give the lengths of time to failure for n = 88 radio transmitter-receivers: a Use the range to approximate s for the n = 88 lengths of time to failure. b Construct a frequency histogram for the data. [Notice the tendency of the distribution to tail outward (skew) to the right.] c Use a calculator (or computer) to calculate y and s. (Hand calculation is much too tedious for this exercise.) d Calculate the intervals yks, k = 1, 2, and 3, and count the number of measurements falling in each interval. Compare your results with the empirical rule results. Note that the empirical rule provides a rather good description of these data, even though the distribution is highly skewed.26SE 27SE The discharge of suspended solids from a phosphate mine is normally distributed with mean daily discharge 27 milligrams per liter (mg/L) and standard deviation 14 mg/L. In what proportion of the days will the daily discharge be less than 13 mg/L?29SE Compared to their stay-at-home peers, women employed outside the home have higher levels of high-density lipoproteins (HDL), the good cholesterol associated with lower risk for heart attacks. A study of cholesterol levels in 2000 women, aged 2564, living in Augsburg, Germany, was conducted by Ursula Haertel, Ulrigh Keil, and colleagues2 at the GSF-Medis Institut in Munich. Of these 2000 women, the 48% who worked outside the home had HDL levels that were between 2.5 and 3.6 milligrams per deciliter (mg/dL) higher than the HDL levels of their stay-at-home counterparts. Suppose that the difference in HDL levels is normally distributed, with mean 0 (indicating no difference between the two groups of women) and standard deviation 1.2 mg/dL. If you were to select an employed woman and a stay-at-home counterpart at random, what is the probability that the difference in their HDL levels would be between 1.2 and 2.4?31SE 32SE 33SE 34SE 35SE 36SE 37SE 38SE For each of the following situations, identify the population of interest, the inferential objective, and how you might go about collecting a sample. a A university researcher wants to estimate the proportion of U.S. citizens from Generation X who are interested in starting their own businesses. b For more than a century, normal body temperature for humans has been accepted to be 98.6 Fahrenheit. Is it really? Researchers want to estimate the average temperature of healthy adults in the United States. c A city engineer wants to estimate the average weekly water consumption for single-family dwelling units in the city. d The National Highway Safety Council wants to estimate the proportion of automobile tires with unsafe tread among all tires manufactured by a specific company during the current production year. e A political scientist wants to determine whether a majority of adult residents of a state favor a unicameral legislature. f A medical scientist wants to estimate the average length of time until the recurrence of a certain disease. g An electrical engineer wants to determine whether the average length of life of transistors of a certain type is greater than 500 hours.Are some cities more windy than others? Does Chicago deserve to be nicknamed The Windy City? Given below are the average wind speeds (in miles per hour) for 45 selected U.S. cities: Source: The World Almanac and Book of Facts, 2004. a Construct a relative frequency histogram for these data. (Choose the class boundaries without including the value 35.1 in the range of values.) b The value 35.1 was recorded at Mt. Washington, New Hampshire. Does the geography of that city explain the magnitude of its average wind speed? c The average wind speed for Chicago is 10.3 miles per hour. What percentage of the cities have average wind speeds in excess of Chicagos? d Do you think that Chicago is unusually windy?Of great importance to residents of central Florida is the amount of radioactive material present in the soil of reclaimed phosphate mining areas. Measurements of the amount of 238U in 25 soil samples were as follows (measurements in picocuries per gram): Construct a relative frequency histogram for these data.The top 40 stocks on the over-the-counter (OTC) market, ranked by percentage of outstanding shares traded on one day last year are as follows: a Construct a relative frequency histogram to describe these data. b What proportion of these top 40 stocks traded more than 4% of the outstanding shares? c If one of the stocks is selected at random from the 40 for which the preceding data were taken, what is the probability that it will have traded fewer than 5% of its outstanding shares?Given here is the relative frequency histogram associated with grade point averages (GPAs) of a sample of 30 students: a Which of the GPA categories identified on the horizontal axis are associated with the largest proportion of students? b What proportion of students had GPAs in each of the categories that you identified? c What proportion of the students had GPAs less than 2.65?The relative frequency histogram given next was constructed from data obtained from a random sample of 25 families. Each was asked the number of quarts of milk that had been purchased the previous week. a Use this relative frequency histogram to determine the number of quarts of milk purchased by the largest proportion of the 25 families. The category associated with the largest relative frequency is called the modal category. b What proportion of the 25 families purchased more than 2 quarts of milk? c What proportion purchased more than 0 but fewer than 5 quarts?The self-reported heights of 105 students in a biostatistics class were used to construct the histogram given below. a Describe the shape of the histogram. b Does this histogram have an unusual feature? c Can you think of an explanation for the two peaks in the histogram? Is there some consideration other than height that results in the two separate peaks? What is it?An article in Archaeometry presented an analysis of 26 samples of RomanoBritish pottery, found at four different kiln sites in the United Kingdom. The percentage of aluminum oxide in each of the 26 samples is given below: Source: A. Tubb, A. J. Parker, and G. Nickless, The Analysis of RomanoBritish Pottery by Atomic Absorption Spectrophotometry, Archaeometry 22 (1980): 153. a Construct a relative frequency histogram to describe the aluminum oxide content of all 26 pottery samples. b What unusual feature do you see in this histogram? Looking at the data, can you think of an explanation for this unusual feature?Resting breathing rates for college-age students are approximately normally distributed with mean 12 and standard deviation 2.3 breaths per minute. What fraction of all college-age students have breathing rates in the following intervals? a 9.7 to 14.3 breaths per minute b 7.4 to 16.6 breaths per minute c 9.7 to 16.6 breaths per minute d Less than 5.1 or more than 18.9 breaths per minute It has been projected that the average and standard deviation of the amount of time spent online using the Internet are, respectively, 14 and 17 hours per person per year (many do not use the Internet at all!). a What value is exactly 1 standard deviation below the mean? b If the amount of time spent online using the Internet is approximately normally distributed, what proportion of the users spend an amount of time online that is less than the value you found in part (a)? c Is the amount of time spent online using the Internet approximately normally distributed? Why?The following results on summations will help us in calculating the sample variance s2. For any constant c, a i=1nc=nc. b i=1ncyi=ci=1nyi. c i=1n(xi+yi)=i=1nxi+i=1nyi. Use (a), (b), and (c) to show that s2=1n1i=1n(yiy)2=1n1[i=1nyi21n(i=1nyi)2].12E 13E Refer to Exercise 1.3 and repeat parts (a) and (b) of Exercise 1.13. 1.13 Refer to Exercise 1.2. a Calculate y and s for the data given. b Calculate the interval yks for k = 1, 2, and 3. Count the number of measurements that fall within each interval and compare this result with the number that you would expect according to the empirical rule. 1.3 Of great importance to residents of central Florida is the amount of radioactive material present in the soil of reclaimed phosphate mining areas. Measurements of the amount of 238U in 25 soil samples were as follows (measurements in picocuries per gram): Construct a relative frequency histogram for these data.Refer to Exercise 1.4 and repeat parts (a) and (b) of Exercise 1.13. 1.13 Refer to Exercise 1.2. a Calculate y and s for the data given. b Calculate the interval yks for k = 1, 2, and 3. Count the number of measurements that fall within each interval and compare this result with the number that you would expect according to the empirical rule. 1.4 The top 40 stocks on the over-the-counter (OTC) market, ranked by percentage of outstanding shares traded on one day last year are as follows: a Construct a relative frequency histogram to describe these data. b What proportion of these top 40 stocks traded more than 4% of the outstanding shares? c If one of the stocks is selected at random from the 40 for which the preceding data were taken, what is the probability that it will have traded fewer than 5% of its outstanding shares?16E 17E 18E 19E 20E The manufacturer of a new food additive for beef cattle claims that 80% of the animals fed a diet including this additive should have monthly weight gains in excess of 20 pounds. A large sample of measurements on weight gains for cattle fed this diet exhibits an approximately normal distribution with mean 22 pounds and standard deviation 2 pounds. Do you think the sample information contradicts the manufacturers claim? (Calculate the probability of a weight gain exceeding 20 pounds.)Show that Theorem 2.7 holds for conditional probabilities. That is, if P(B) 0, then P(A|B)=1P(A|B).Let S contain four sample points, E1, E2, E3, and E4. a List all possible events in S (include the null event). b In Exercise 2.68(d), you showed that i=1n(ni)=2n. Use this result to give the total number of events in S. c Let A and B be the events {E1, E2, E3} and {E2, E4}, respectively. Give the sample points in the following events: AB, A B, AB, and AB.145SE 146SE 147SE A bin contains three components from supplier A, four from supplier B, and five from supplier C. If four of the components are randomly selected for testing, what is the probability that each supplier would have at least one component tested?A large group of people is to be checked for two common symptoms of a certain disease. It is thought that 20% of the people possess symptom A alone, 30% possess symptom B alone, 10%possess both symptoms, and the remainder have neither symptom. For one person chosen at random from this group, find these probabilities: a The person has neither symptom. b The person has at least one symptom. c The person has both symptoms, given that he has symptom B.2.149 A large group of people is to be checked for two common symptoms of a certain disease. It is thought that 20% of the people possess symptom A alone, 30% possess symptom B alone, 10%possess both symptoms, and the remainder have neither symptom. For one person chosen at random from this group, find these probabilities: a The person has neither symptom. b The person has at least one symptom. c The person has both symptoms, given that he has symptom B. Refer to Exercise 2.149. Let the random variable Y represent the number of symptoms possessed by a person chosen at random from the group. Compute the probabilities associated with each value of Y.151SE 152SE 153SE a A drawer contains n = 5 different and distinguishable pairs of socks (a total of ten socks). If a person (perhaps in the dark) randomly selects four socks, what is the probability that there is no matching pair in the sample? b A drawer contains n different and distinguishable pairs of socks (a total of 2n socks). A person randomly selects 2r of the socks, where 2r n. In terms of n and r, what is the probability that there is no matching pair in the sample?A group of men possesses the three characteristics of being married (A), having a college degree (B), and being a citizen of a specified state (C), according to the fractions given in the accompanying Venn diagram. That is, 5% of the men possess all three characteristics, whereas 20% have a college education but are not married and are not citizens of the specified state. One man is chosen at random from this group. Find the probability that he a is married. b has a college degree and is married. c is not from the specified state but is married and has a college degree. d is not married or does not have a college degree, given that he is from the specified state.The accompanying table lists accidental deaths by age and certain specific types for the United States in 2002. a A randomly selected person from the United States was known to have an accidental death in 2002. Find the probability that i. he was over the age of 15 years. ii. the cause of death was a motor vehicle accident. iii. the cause of death was a motor vehicle accident, given that the person was between 15 and 24 years old. iv. the cause of death was a drowning accident, given that it was not a motor vehicle accident and the person was 34 years old or younger. b From these figures can you determine the probability that a person selected at random from the U.S. population had a fatal motor vehicle accident in 2002? Source: Compiled from National Vital Statistics Report 50, no. 15, 2002.157SE A bowl contains w white balls and b black balls. One ball is selected at random from the bowl, its color is noted, and it is returned to the bowl along with n additional balls of the same color. Another single ball is randomly selected from the bowl (now containing w + b + n balls) and it is observed that the ball is black. Show that the (conditional) probability that the first ball selected was white is ww+b+n.159SE A machine for producing a new experimental electronic component generates defectives from time to time in a random manner. The supervising engineer for a particular machine has noticed that defectives seem to be grouping (hence appearing in a nonrandom manner), thereby suggesting a malfunction in some part of the machine. One test for nonrandomness is based on the number of runs of defectives and nondefectives (a run is an unbroken sequence of either defectives or nondefectives). The smaller the number of runs, the greater will be the amount of evidence indicating nonrandomness. Of 12 components drawn from the machine, the first 10 were not defective, and the last 2 were defective (NNNNNNNNNNDD). Assume randomness. What is the probability of observing a this arrangement (resulting in two runs) given that 10 of the 12 components are not defective? b two runs?161SE 162SE Relays used in the construction of electric circuits function properly with probability .9. Assuming that the circuits operate independently, which of the following circuit designs yields the higher probability that current will flow when the relays are activated?164SE Refer to Exercise 2.163 and consider circuit B. If we know that current is flowing, what is the probability that switches 1 and 4 are functioning properly? 2.163 Relays used in the construction of electric circuits function properly with probability .9. Assuming that the circuits operate independently, which of the following circuit designs yields the higher probability that current will flow when the relays are activated?166SE 167SE As in Exercises 2.166 and 2.167, eight tires of different brands are ranked from 1 to 8 (best to worst) according to mileage performance. a If four of these tires are chosen at random by a customer, what is the probability that the best tire selected is ranked 3 and the worst is ranked 7? b In part (a) you computed the probability that the best tire selected is ranked 3 and the worst is ranked 7. If that is the case, the range of the ranks, R = largest rank smallest rank = 7 3 = 4. What is P(R = 4)? c Give all possible values for R and the probabilities associated with all of these values. 2.167 Refer to Exercise 2.166. Let Y denote the actual quality rank of the best tire selected by the customer. In Exercise 2.166, you computed P(Y = 3). Give the possible values of Y and the probabilities associated with all of these values. 2.166 Eight tires of different brands are ranked from 1 to 8 (best to worst) according to mileage performance. If four of these tires are chosen at random by a customer, find the probability that the best tire among those selected by the customer is actually ranked third among the original eight.Three beer drinkers (say I, II, and III) are to rank four different brands of beer (say A, B, C, and D) in a blindfold test. Each drinker ranks the four beers as 1 (for the beer that he or she liked best), 2 (for the next best), 3, or 4. a Carefully describe a sample space for this experiment (note that we need to specify the ranking of all four beers for all three drinkers). How many sample points are in this sample space? b Assume that the drinkers cannot discriminate between the beers so that each assignment of ranks to the beers is equally likely. After all the beers are ranked by all three drinkers, the ranks of each brand of beer are summed. What is the probability that some beer will receive a total rank of 4 or less?170SE 171SE 172SE 173SE 174SE 175SE 176SE Refer to Exercise 2.90(b) where a friend claimed that if there is a 1 in 50 chance of injury on a single jump then there is a 100% chance of injury if a skydiver jumps 50 times. Assume that the results of repeated jumps are mutually independent. a. What is the probability that 50 jumps will be completed without an injury? b. What is the probability that at least one injury will occur in 50 jumps? c. What is the maximum number of jumps, n. the skydiver can make if the probability is at least .60 that all n jumps will be completed without injury? 2.90 Suppose that there is a 1 in 50 chance of injury on a single skydiving attempt. a. If we assume that the outcomes of different jumps are independent, what is the probability that a skydiver is injured if she jumps twice? b. A friend claims if there is a 1 in 50 chance of injury on a single jump then there is a 100% chance of injury if a skydiver jumps 50 times. Is your friend correct? Why?Suppose that the probability of exposure to the flu during an epidemic is .6. Experience has shown that a serum is 80% successful in preventing an inoculated person from acquiring the flu, if exposed to it. A person not inoculated faces a probability of .90 of acquiring the flu if exposed to it. Two persons, one inoculated and one not, perform a highly specialized task in a business. Assume that they are not at the same location, are not in contact with the same people, and cannot expose each other to the flu. What is the probability that at least one will get the flu?Two gamblers bet 1 each on the successive tosses of a coin. Each has a bank of 6. What is the probability that a. they break even after six tosses of the coin? b. one playersay, Joneswins all the money on the tenth toss of the coin?180SE Suppose that n indistinguishable balls are to be arranged in N distinguishable boxes so that each distinguishable arrangement is equally likely. If n N, show that the probability no box will be empty is given by (n1N1)(N+n1N1).Suppose a family contains two children of different ages, and we are interested in the gender of these children. Let F denote that a child is female and M that the child is male and let a pair such as FM denote that the older child is female and the younger is male. There are four points in the set S of possible observations: S={FF,FM,MF,MM}. Let A denote the subset of possibilities containing no males; B, the subset containing two males; and C, the subset containing at least one male. List the elements of A, B, C, A B, A B, A C, A C, B C, B C, and CB.Suppose that A and B are two events. Write expressions involving unions, intersections, and complements that describe the following: a Both events occur. b At least one occurs. c Neither occurs. d Exactly one occurs.3E 4E Refer to Exercise 2.4. Use the identities A=ASandS=BB and a distributive law to prove that a A=(AB)(AB). b If BA then A=B(AB). c Further, show that (AB) and (AB) are mutually exclusive and therefore that A is the union of two mutually exclusive sets, (AB) and (AB). d Also show that B and (AB) are mutually exclusive and if BA, A is the union of two mutually exclusive sets. B and (AB).Suppose two dice are tossed and the numbers on the upper faces are observed. Let S denote the set of all possible pairs that can be observed. [These pairs can be listed, for example, by letting (2, 3) denote that a 2 was observed on the first die and a 3 on the second.] a Define the following subsets of S: A: The number on the second die is even. B: The sum of the two numbers is even. C: At least one number in the pair is odd. b List the points in A,C,AB,AB,AB,andAC.A group of five applicants for a pair of identical jobs consists of three men and two women. The employer is to select two of the five applicants for the jobs. Let S denote the set of all possible outcomes for the employers selection. Let A denote the subset of outcomes corresponding to the selection of two men and B the subset corresponding to the selection of at least one woman. List the outcomes in A,B,AB,AB,andAB. (Denote the different men and women by M1, M2, M3 and W1, W2, respectively.)From a survey of 60 students attending a university, it was found that 9 were living off campus. 36 were undergraduates, and 3 were undergraduates living off campus. Find the number of these students who were a undergraduates, were living off campus, or both. b undergraduates living on campus. c graduate students living on campus.The proportions of blood phenotypes. A, B, AB, and O, in the population of all Caucasians in the United States are approximately .41, .10, .04, and .45, respectively. A single Caucasian is chosen at random from the population. a List the sample space for this experiment. b Make use of the information given above to assign probabilities to each of the simple events. c What is the probability that the person chosen at random has either type A or type AB blood?The proportions of blood phenotypes. A, B, AB, and O, in the population of all Caucasians in the United States are approximately .41, .10, .04, and .45, respectively. A single Caucasian is chosen at random from the population. a List the sample space for this experiment. b Make use of the information given above to assign probabilities to each of the simple events. c What is the probability that the person chosen at random has either type A or type AB blood?A sample space consists of five simple events. E1, E2, E3, E4, and E5. a If P(E1) = P(E2) = 0.15, P(E3) = 0.4, and P(E4) = 2P(E5), find the probabilities of E4 and E5. b If P(E1) = 3P(E2) = 0.3, find the probabilities of the remaining simple events if you know that the remaining simple events are equally probable.A vehicle arriving at an intersection can turn right, turn left, or continue straight ahead. The experiment consists of observing the movement of a single vehicle through the intersection. a List the sample space for this experiment. b Assuming that all sample points are equally likely, find the probability that the vehicle turns.Americans can be quite suspicious, especially when it comes to government conspiracies. On the question of whether the U.S. Air Force has withheld proof of the existence of intelligent life on other planets, the proportions of Americans with varying opinions arc given in the table. Suppose that one American is selected and his or her opinion is recorded. a What are the simple events for this experiment? b Are the simple events that you gave in part (a) all equally likely? If not, what are the probabilities that should be assigned to each? c What is the probability that the person selected finds it at least somewhat likely that the Air Force is withholding information about intelligent life on other planets?A survey classified a large number of adults according to whether they were diagnosed as needing eyeglasses to correct their reading vision and whether they use eyeglasses when reading. The proportions falling into the four resulting categories are given in the following table: If a single adult is selected from the large group, find the probabilities of the events defined below. The adult a needs glasses. b needs glasses but does not use them. c uses glasses whether the glasses are needed or not.An oil prospecting firm hits oil or gas on 10% of its drillings. If the firm drills two wells, the four possible simple events and three of their associated probabilities are as given in the accompanying table, bind the probability that the company will hit oil or gas a on the first drilling and miss on the second. b on at least one of the two drillings.16E Hydraulic landing assemblies coming from an aircraft rework facility are each inspected for defects. Historical records indicate that 8% have defects in shafts only, 6% have defects in bushings only, and 2% have defects in both shafts and bushings. One of the hydraulic assemblies is selected randomly. What is the probability that the assembly has a a bushing defect? b a shaft or bushing defect? c exactly one of the two types of defects? d neither type of defect?Suppose two balanced coins are tossed and the upper faces are observed. a List the sample points for this experiment. b Assign a reasonable probability to each sample point. (Are the sample points equally likely?) c Let A denote the event that exactly one head is observed and B the event that at least one head is observed. List the sample points in both A and B. d From your answer to part (c), find P(A), P(B), P(A B), P(A B), and P(AB).A business office orders paper supplies from one of three vendors, V1, V2, or V3. Orders are to be placed on two successive days, one order per day. Thus, (V2, V3) might denote that vendor V2 gets the order on the first day and vendor V3 gets the order on the second day. a List the sample points in this experiment of ordering paper on two successive days. b Assume the vendors are selected at random each day and assign a probability to each sample point. c Let A denote the event that the same vendor gets both orders and B the event that V2 gets at least one order. Find P(A), P(B), P(A B), and P(A B) by summing the probabilities of the sample points in these events.The following game was played on a popular television show. The host showed a contestant three large curtains. Behind one of the curtains was a nice prize (maybe a new car) and behind the other two curtains were worthless prizes (duds). The contestant was asked to choose one curtain, If the curtains are identified by their prizes, they could be labeled G, D1, and D2 (Good Prize, Dud1.and Dud2). Thus, the sample space for the contestants choice is S = {G, D1, D2).1 a If the contestant has no idea which curtains hide the various prizes and selects a curtain at random, assign reasonable probabilities to the simple events and calculate the probability that the contestant selects the curtain hiding the nice prize. b Before showing the contestant what was behind the curtain initially chosen, the game show host would open one of the curtains and show the contestant one of the duds (he could always do this because he knew the curtain hiding the good prize). He then offered the contestant the option of changing from the curtain initially selected to the other remaining unopened curtain. Which strategy maximizes the contestants probability of winning the good prize: stay with the initial choice or switch to the other curtain? In answering the following sequence of questions, you will discover that, perhaps surprisingly, this question can be answered by considering only the sample space above and using the probabilities that you assigned to answer part (a). i If the contestant choses to stay with her initial choice, she wins the good prize if and only if she initially chose curtain G. If she stays with her initial choice, what is the probability that she wins the good prize? ii If the host shows her one of the duds and she switches to the other unopened curtain, what will be the result if she had initially selected G? iii Answer the question in part (ii) if she had initially selected one of the duds. iv If the contestant switches from her initial choice (as the result of being shown one of the duds), what is the probability that the contestant wins the good prize? v Which strategy maximizes the contestants probability of winning the good prize: stay with the initial choice or switch to the other curtain?If A and B are events, use the result derived in Exercise 2.5(a) and the Axioms in Definition 2.6 to prove that P(A)=P(AB)+P(AB).If A and B are events and B A, use the result derived in Exercise 2.5(b) and the Axioms in Definition 2.6 to prove that P(A)=P(B)+P(AB).23E Use the result in Exercise 2.22 and the Axioms in Definition 2.6 to prove the obvious result in Exercise 2.23.A single car is randomly selected from among all of those registered at a local tag agency. What do you think of the following claim? All cars are either Volkswagens or they are not. Therefore, the probability is 1/2 that the car selected is a Volkswagen.According to Websters New Collegiate Dictionary, a divining rod is a forked rod believed to indicate (divine) the presence of water or minerals by dipping downward when held over a vein. To test the claims of a divining rod expert, skeptics bury four cans in the ground, two empty and two filled with water. The expert is led to the four cans and told that two contain water. He uses the divining rod to test each of the four cans and decide which two contain water. a List the sample space for this experiment. b If the divining rod is completely useless for locating water, what is the probability that the expert will correctly identify (by guessing) both of the cans containing water?In Exercise 2.12 we considered a situation where cars entering an intersection each could turn right, turn left, or go straight. An experiment consists of observing two vehicles moving through the intersection. a How many sample points are there in the sample space? List them. b Assuming that all sample points are equally likely, what is the probability that at least one car turns left? c Again assuming equally likely sample points, what is the probability that at most one vehicle turns? 2.12 A vehicle arriving at an intersection can turn right, turn left, or continue straight ahead. The experiment consists of observing the movement of a single vehicle through the intersection. a List the sample space for this experiment. b Assuming that all sample points are equally likely, find the probability that the vehicle turns.Four equally qualified people apply for two identical positions in a company. One and only one applicant is a member of a minority group. The positions are filled by choosing two of the applicants at random. a List the possible outcomes for this experiment. b Assign reasonable probabilities to the sample points. c Find the probability that the applicant from the minority group is selected for a position.Two additional jurors are needed to complete a jury for a criminal trial. There are six prospective jurors, two women and four men. Two jurors are randomly selected from the six available. a Define the experiment and describe one sample point. Assume that you need describe only the two jurors chosen and not the order in which they were selected. b List the sample space associated with this experiment. c What is the probability that both of the jurors selected are women?Three imported wines are to be ranked from lowest to highest by a purported wine expert. That is, one wine will be identified as best, another as second best, and the remaining wine as worst. a Describe one sample point for this experiment. b List the sample space. c Assume that the expert really knows nothing about wine and randomly assigns ranks to the three wines. One of the wines is of much better quality than the others. What is the probability that the expert ranks the best wine no worse than second best?A boxcar contains six complex electronic systems. Two of the six are to be randomly selected for thorough testing and then classified as defective or not defective. a If two of the six systems are actually defective, find the probability that at least one of the two systems tested will be defective. Find the probability that both are defective. b If four of the six systems are actually defective, find the probabilities indicated in part (a).A retailer sells only two styles of stereo consoles, and experience shows that these are in equal demand. Four customers in succession come into the store to order stereos. The retailer is interested in their preferences. a List the possibilities for preference arrangements among the four customers (that is, list the sample space). b Assign probabilities to the sample points. c Let A denote the event that all four customers prefer the same style. Find P(A).The Bureau of the Census reports that the median family income for all families in the United States during the year 2003 was 43,318. That is, half of all American families had incomes exceeding this amount, and half had incomes equal to or below this amount. Suppose that four families are surveyed and that each one reveals whether its income exceeded 43,318 in 2003. a List the points in the sample space. b Identify the simple events in each of the following events: A: At least two had incomes exceeding 43,318. B: Exactly two had incomes exceeding 43,318. C: Exactly one had income less than or equal to 43,318. c Make use of the given interpretation for the median to assign probabilities to the simple events and find P(A), P(B), and P(C).Patients arriving at a hospital outpatient clinic can select one of three stations for service. Suppose that physicians are assigned randomly to the stations and that the patients therefore have no station preference. Three patients arrive at the clinic and their selection of stations is observed. a List the sample points for the experiment. b Let A be the event that each station receives a patient. List the sample points in A. c Make a reasonable assignment of probabilities to the sample points and find P(A).An airline has six flights from New York to California and seven flights from California to Hawaii per day. If the flights are to be made on separate days, how many different flight arrangements can the airline offer from New York to Hawaii?36E A businesswoman in Philadelphia is preparing an itinerary for a visit to six major cities. The distance traveled, and hence the cost of the trip, will depend on the order in which she plans her route. a How many different itineraries (and trip costs) are possible? b If the businesswoman randomly selects one of the possible itineraries and Denver and San Francisco are two of the cities that she plans to visit, what is the probability that she will visit Denver before San Francisco?An upscale restaurant offers a special fixe prix menu in which, for a fixed dinner cost, a diner can select from four appetizers, three salads, four entrees, and five desserts. How many different dinners are available if a dinner consists of one appetizer, one salad, one entree, and one dessert?An experiment consists of tossing a pair of dice. a Use the combinatorial theorems to determine the number of sample points in the sample space S. b Find the probability that the sum of the numbers appearing on the dice is equal to 7.A brand of automobile comes in five different styles, with four types of engines, with two types of transmissions, and in eight colors. a How many autos would a dealer have to stock if he included one for each styleenginetransmission combination? b How many would a distribution center have to carry if all colors of ears were stocked for each combination in part (a)?41E 42E A fleet of nine taxis is to be dispatched to three airports in such a way that three go to airport A, five go to airport B, and one goes to airport C. In how many distinct ways can this be accomplished?Refer to Exercise 2.43. Assume that taxis are allocated to airports at random. a If exactly one of the taxis is in need of repair, what is the probability that it is dispatched to airport C? b If exactly three of the taxis are in need of repair, what is the probability that every airport receives one of the taxis requiring repairs? 2.43 A fleet of nine taxis is to he dispatched to three airports in such a way that three go to airport A, five go to airport B, and one goes to airport C. In how many distinct ways can this he accomplished?45E Ten teams are playing in a basketball tournament. In the first round, the teams are randomly assigned to games 1, 2, 3, 4 and 5. In how many ways can the teams be assigned to the games?Refer to Exercise 2.46. If 2n teams are to be assigned to games 1, 2, , n, in how many ways can the teams be assigned to the games? 2.46 Ten teams are playing in a basketball tournament. In the first round, the teams are randomly assigned to games 1, 2, 3, 4 and 5. In how many ways can the teams be assigned to the games?48E Students attending the University of Florida can select from 130 major areas of study. A students major is identified in the registrars records with a two-or three-letter code (for example, statistics majors are identified by STA, math majors by MS). Some students opt for a double major and complete the requirements for both of the major areas before graduation. The registrar was asked to consider assigning these double majors a distinct two- or three-letter code so that they could be identified through the student records system. a What is the maximum number of possible double majors available to University of Florida students? b If any two- or three-letter code is available to identify majors or double majors, how many major codes are available? c How many major codes are required to identify students who have either a single major or a double major? d Are there enough major codes available to identify all single and double majors at the University of Florida?50E A local fraternity is conducting a raffle where 50 tickets are to be soldone per customer. There are three prizes to be awarded. If the four organizers of the raffle each buy one ticket, what is the probability that the four organizers win a all of the prizes? b exactly two of the prizes? c exactly one of the prizes? d none of the prizes?An experimenter wishes to investigate the effect of three variablespressure, temperature, and the type of catalyston the yield in a refining process. If the experimenter intends to use three settings each for temperature and pressure and two types of catalysts, how many experimental runs will have to be conducted if he wishes to run all possible combinations of pressure, temperature, and types of catalysts?Five firms, F1, F2,, F5, each offer bids on three separate contracts, C1, C2, and C3. Any one firm will be awarded at most one contract. The contracts are quite different, so an assignment of C1 to F1, say, is to be distinguished from an assignment of C2 to F1. a How many sample points are there altogether in this experiment involving assignment of contracts to the firms? (No need to list them all.) b Under the assumption of equally likely sample points, find the probability that F3 is awarded a contract.A group of three undergraduate and five graduate students are available to fill certain student government posts. If four students are to be randomly selected from this group, find the probability that exactly two undergraduates will be among the four chosen.A study is to be conducted in a hospital to determine the attitudes of nurses toward various administrative procedures. A sample of 10 nurses is to be selected from a total of the 90 nurses employed by the hospital. a How many different samples of 10 nurses can be selected? b Twenty of the 90 nurses are male. If 10 nurses are randomly selected from those employed by the hospital, what is the probability that the sample of ten will include exactly 4 male (and 6 female) nurses?A student prepares for an exam by studying a list of ten problems. She can solve six of them. For the exam, the instructor selects five problems at random from the ten on the list given to the students. What is the probability that the student can solve all five problems on the exam?Two cards are drawn from a standard 52-card playing deck. What is the probability that the draw will yield an ace and a face card?Five cards are dealt from a standard 52-card deck. What is the probability that we draw a 3 aces and 2 kings? b a full house (3 cards of one kind, 2 cards of another kind)?59E Refer to Example 2.7. Suppose that we record the birthday for each of n randomly selected persons. a Give an expression for the probability that none share the same birthday. b What is the smallest value of n so that the probability is at least .5 that at least two people share a birthday?61E A manufacturer has nine distinct motors in stock, two of which came from a particular supplier. The motors must be divided among three production lines, with three motors going to each line. If the assignment of motors to lines is random, find the probability that both motors from the particular supplier are assigned to the first line.The eight-member Human Relations Advisory Board of Gainesville, Florida, considered the complaint of a woman who claimed discrimination, based on sex, on the part of a local company. The board, composed of five women and three men, voted 53 in favor of the plaintiff, the five women voting in favor of the plaintiff, the three men against. The attorney representing the company appealed the boards decision by claiming sex bias on the part of the board members. If there was no sex bias among the board members, it might be reasonable to conjecture that any group of five board members would be as likely to vote for the complainant as any other group of five. If this were the case, what is the probability that the vote would split along sex lines (five women for, three men against)?A balanced die is tossed six times, and the number on the uppermost face is recorded each time. What is the probability that the numbers recorded are 1, 2, 3, 4, 5, and 6 in any order?65E Refer to Example 2.10. What is the probability that a an ethnic group member is assigned to each type of job? b no ethnic group member is assigned to a type 4 job? EXAMPLE 2.10 A labor dispute has arisen concerning the distribution of 20 laborers to four different construction jobs. The first job (considered to be very undesirable) required 6 laborers; the second, third, and fourth utilized 4, 5, and 5 laborers, respectively. The dispute arose over an alleged random distribution of the laborers to the jobs that placed all 4 members of a particular ethnic group on job 1. In considering whether the assignment represented injustice, a mediation panel desired the probability of the observed event. Determine the number of sample points in the sample space S for this experiment. That is, determine the number of ways the 20 laborers can be divided into groups of the appropriate sizes to fill all of the jobs. Find the probability of the observed event if it is assumed that the laborers are randomly assigned to jobs.Refer to Example 2.13. Suppose that the number of distributors is M = 10 and that there are n = 7 orders to be placed. What is the probability that a all of the orders go to different distributors? b distributor I gets exactly two orders and distributor II gets exactly three orders? c distributors I, II, and III get exactly two, three, and one order(s), respectively? EXAMPLE 2.13 A company orders supplies from M distributors and wishes to place n orders (n M). Assume that the company places the orders in a manner that allows every distributor an equal chance of obtaining any one order and there is no restriction on the number of orders that can be placed with any distributor. Find the probability that a particular distributorsay, distributor Igets exactly k orders (k n).Show that, for any integer n 1, a (nn)=1. Interpret this result. b (n0)=1 Interpret this result. c (nr)=(nnr). Interpret this result. d i=0n(ni)=2n. [Hint: Consider the binomial expansion of (x + y)n with x = y = 1.]69E 70E If two events, A and B, are such that P(A) = .5, P(B) = .3, and P(A B) = .1, find the following: a P(A|B) b P(B|A) c P(A|AB) d P(A|AB) e P(AB|AB)For a certain population of employees, the percentage passing or failing a job competency exam, listed according to sex, were as shown in the accompanying table. That is, of all the people taking the exam, 24% were in the male-pass category, 16% were in the male-fail category, and so forth. An employee is to be selected randomly from this population. Let A be the event that the employee scores a passing grade on the exam and let M be the event that a male is selected. a Are the events A and M independent? b Are the events A and F independent?Gregor Mendel was a monk who, in 1865, suggested a theory of inheritance based on the science of genetics. He identified heterozygous individuals for flower color that had two alleles (one r = recessive white color allele and one R = dominant red color allele). When these individuals were mated, 3/4 of the offspring were observed to have red flowers, and 1/4 had white flowers. The following table summarizes this mating; each parent gives one of its alleles to form the gene of the offspring. We assume that each parent is equally likely to give either of the two alleles and that, if either one or two of the alleles in a pair is dominant (R), the offspring will have red flowers. What is the probability that an offspring has a at least one dominant allele? b at least one recessive allele? c one recessive allele, given that the offspring has red flowers?One hundred adults were interviewed in a telephone survey. Of interest was their opinions regarding the loan burdens of college students and whether the respondent had a child currently in college. Their responses are summarized in the table below: Which of the following are independent events? a A and D b B and D c C and D 75E A survey of consumers in a particular community showed that 10% were dissatisfied with plumbing jobs done in their homes. Half the complaints dealt with plumber A, who does 40% of the plumbing jobs in the town. Find the probability that a consumer will obtain a an unsatisfactory plumbing job, given that the plumber was A. b a satisfactory plumbing job, given that the plumber was A.A study of the posttreatment behavior of a large number of drug abusers suggests that the likelihood of conviction within a two-year period after treatment may depend upon the offenders education. The proportions of the total number of cases falling in four educationconviction categories are shown in the following table: Suppose that a single offender is selected from the treatment program. Define the events: A: The offender has 10 or more years of education. B: The offender is convicted within two years after completion of treatment. Find the following: a P(A). b P(B). c P(A B). d P(A B). e P(A). f P(AB). g P(AB). h P(A|B). i P(B|A).In the definition of the independence of two events, you were given three equalities to check: P(A|B) = P(A) or P(B|A) = P(B) or P(A B) = P(A)P(B). If any one of these equalities holds, A and B are independent. Show that if any of these equalities hold, the other two also hold.Suppose that A and B are mutually exclusive events, with P(A) 0 and P(B) 1. Are A and B independent? Prove your answer.Suppose that A B and that P(A) 0 and P(B) 0. Are A and B independent? Prove your answer.If P(A) 0, P(B) 0, and P(A) P(A|B), show that P(B) P(B|A).Suppose that A B and that P(A) 0 and P(B) 0. Show that P(B|A) = 1 and P(A|B) = P(A)/P(B).If A and B are mutually exclusive events and P(B) 0, show that P(A|AB)=P(A)P(A)+P(B).If A1, A2, and A3 are three events and P(A1A2)=P(A1A3)0 but P(A2 A3) = 0, show that P(atleastoneAi)=P(A1)+P(A2)+P(A3)2P(A1A2).85E Suppose that A and B are two events such that P(A) = .8 and P(B) = .7. a Is it possible that P(A B) = .1? Why or why not? b What is the smallest possible value for P(A B)? c Is it possible that P(A B) = .77? Why or why not? d What is the largest possible value for P(A B)?Suppose that A and B are two events such that P(A) + P(B) 1. a What is the smallest possible value for P(A B)? b What is the largest possible value for P(A B)?Suppose that A and B are two events such that P(A) = .6 and P( B) = .3. a Is it possible that P(A B) = .1? Why or why not? b What is the smallest possible value for P(A B)? c Is it possible that P(A B) = .7? Why or why not? d What is the largest possible value for P(A B)?89E Suppose that there is a 1 in 50 chance of injury on a single skydiving attempt. a If we assume that the outcomes of different jumps are independent, what is the probability that a skydiver is injured if she jumps twice? b A friend claims if there is a 1 in 50 chance of injury on a single jump then there is a 100% chance of injury if a skydiver jumps 50 times. Is your friend correct? Why?Can A an B be mutually exclusive if P(A) = .4 and P(B) = .7? If P(A) = .4 and P(B) = .3? Why?92E In a game, a participant is given three attempts to hit a ball. On each try, she either scores a hit, H, or a miss, M. The game requires that the player must alternate which hand she uses in successive attempts. That is, if she makes her first attempt with her right hand, she must use her left hand for the second attempt and her right hand for the third. Her chance of scoring a hit with her right hand is .7 and with her left hand is .4. Assume that the results of successive attempts are independent and that she wins the game if she scores at least two hits in a row. If she makes her first attempt with her right hand, what is the probability that she wins the game?A smoke detector system uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is .95; by device B, .90; and by both devices, .88. a If smoke is present, find the probability that the smoke will be detected by either device A or B or both devices. b Find the probability that the smoke will be undetected.95E If A and B are independent events with P(A) = .5 and P(B) = .2, find the following: a P(AB) b P(AB) c P(AB)Consider the following portion of an electric circuit with three relays. Current will flow from point a to point b if there is at least one closed path when the relays are activated. The relays may malfunction and not close when activated. Suppose that the relays act independently of one another and close properly when activated, with a probability of .9. a What is the probability that current will flow when the relays are activated? b Given that current flowed when the relays were activated, what is the probability that relay 1 functioned?With relays operating as in Exercise 2.97, compare the probability of current flowing from a to b in the series system shown with the probability of flow in the parallel system shown. 2.97 Consider the following portion of an electric circuit with three relays. Current will flow from point a to point b if there is at least one closed path when the relays are activated. The relays may malfunction and not close when activated. Suppose that the relays act independently of one another and close properly when activated, with a probability of .9. a What is the probability that current will flow when the relays are activated? b Given that current flowed when the relays were activated, what is the probability that relay 1 functioned?Suppose that A and B are independent events such that the probability that neither occurs is a and the probability of B is b. Show that P(A)=1ba1b.Show that Theorem 2.6, the additive law of probability, holds for conditional probabilities. That is, if A, B, and C are events such that P(C) 0, prove that P(AB|C)=P(A|C)+P(B|C)P(AB|C). [Hint: Make use of the distributive law (AB)C=(AC)(BC).]Articles coming through an inspection line are visually inspected by two successive inspectors. When a defective article comes through the inspection line, the probability that it gets by the first inspector is .1. The second inspector will miss five out of ten of the defective items that get past the first inspector. What is the probability that a defective item gets by both inspectors?Diseases I and II are prevalent among people in a certain population. It is assumed that 10% of the population will contract disease I sometime during their lifetime, 15% will contract disease II eventually, and 3%will contract both diseases. a Find the probability that a randomly chosen person from this population will contract at least one disease. b Find the conditional probability that a randomly chosen person from this population will contract both diseases, given that he or she has contracted at least one disease.Refer to Exercise 2.50. Hours after the rigging of the Pennsylvania state lottery was announced, Connecticut state lottery officials were stunned to learn that their winning number for the day was 666 (Los Angeles Times, September 21, 1980). a All evidence indicates that the Connecticut selection of 666 was due to pure chance. What is the probability that a 666 would be drawn in Connecticut, given that a 666 had been selected in the April 24, 1980, Pennsylvania lottery? b What is the probability of drawing a 666 in the April 24, 1980, Pennsylvania lottery (remember, this drawing was rigged) and a 666 in the September 19, 1980, Connecticut lottery? 2.50 Probability played a role in the rigging of the April 24, 1980, Pennsylvania state lottery (Los Angeles Times, September 8, 1980). To determine each digit of the three-digit winning number, each of the numbers 0, 1, 2, , 9 is placed on a Ping-Pong ball, the ten balls are blown into a compartment, and the number selected for the digit is the one on the ball that floats to the top of the machine. To alter the odds, the conspirators injected a liquid into all balls used in the game except those numbered 4 and 6, making it almost certain that the lighter balls would be selected and determine the digits in the winning number. Then they bought lottery tickets bearing the potential winning numbers. How many potential winning numbers were there (666 was the eventual winner)?If A and B are two events, prove that P(AB)1P(A)P(B). [Note: This is a simplified version of the Bonferroni inequality.]If the probability of injury on each individual parachute jump is .05, use the result in Exercise 2.104 to provide a lower bound for the probability of landing safely on both of two jumps.If A and B are equally likely events and we require that the probability of their intersection be at least .98, what is P(A)?107E If A, B, and C are three events, use two applications of the result in Exercise 2.104 to prove that P(ABC)1P(A)P(B)P(C).109E Of the items produced daily by a factory, 40% come from line I and 60% from line II. Line I has a defect rate of 8%, whereas line II has a defect rate of 10%. If an item is chosen at random from the days production, find the probability that it will not be defective.111E Three radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each set has a probability of .02 of failing to detect a plane in its area. If an aircraft enters the area, what is the probability that it a goes undetected? b is detected by all three radar sets?Consider one of the radar sets of Exercise 2.112. What is the probability that it will correctly detect exactly three aircraft before it fails to detect one, if aircraft arrivals are independent single events occurring at different times? 2.112 Three radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each set has a probability of .02 of failing to detect a plane in its area. If an aircraft enters the area, what is the probability that it a goes undetected? b is detected by all three radar sets?A lie detector will show a positive reading (indicate a lie) 10% of the time when a person is telling the truth and 95% of the time when the person is lying. Suppose two people are suspects in a one-person crime and (for certain) one is guilty and will lie. Assume further that the lie detector operates independently for the truthful person and the liar. What is the probability that the detector a shows a positive reading for both suspects? b shows a positive reading for the guilty suspect and a negative reading for the innocent suspect? c is completely wrongthat is, that it gives a positive reading for the innocent suspect and a negative reading for the guilty? d gives a positive reading for either or both of the two suspects?115E A communications network has a built-in safeguard system against failures. In this system if line I fails, it is bypassed and line II is used. If line II also fails, it is bypassed and line III is used. The probability of failure of any one of these three lines is .01, and the failures of these lines are independent events. What is the probability that this system of three lines does not completely fail?A slate auto-inspection station has two inspection teams. Team 1 is lenient and passes all automobiles of a recent vintage; team 2 rejects all autos on a first inspection because their headlights are not properly adjusted. Four unsuspecting drivers take their autos to the station for inspection on four different days and randomly select one of the two teams. a If all four cars are new and in excellent condition, what is the probability that three of the four will be rejected? b What is the probability that all four will pass?118E Suppose that two balanced dice are tossed repeatedly and the sum of the two uppermost faces is determined on each toss. What is the probability that we obtain a a sum of 3 before we obtain a sum of 7? b a sum of 4 before we obtain a sum of 7?Suppose that two defective refrigerators have been included in a shipment of six refrigerators. The buyer begins to test the six refrigerators one at a time. a What is the probability that the last defective refrigerator is found on the fourth test? b What is the probability that no more than four refrigerators need to be tested to locate both of the defective refrigerators? c When given that exactly one of the two defective refrigerators has been located in the first two tests, what is the probability that the remaining defective refrigerator is found in the third or fourth test?121E Applet Exercise Use the applet Bayes Rule as a Tree to obtain the results given in Figure 2.13.123E A population of voters contains 40% Republicans and 60% Democrats. It is reported that 30% of the Republicans and 70% of the Democrats favor an election issue. A person chosen at random from this population is found to favor the issue in question. Find the conditional probability that this person is a Democrat.A diagnostic test for a disease is such that it (correctly) detects the disease in 90% of the individuals who actually have the disease. Also, if a person does not have the disease, the test will report that he or she does not have it with probability .9. Only 1% of the population has the disease in question. If a person is chosen at random from the population and the diagnostic test indicates that she has the disease, what is the conditional probability that she does, in fact, have the disease? Are you surprised by the answer? Would you call this diagnostic test reliable?126E 127E Use Theorem 2.8, the law of total probability, to prove the following: a If P(A|B)=P(A|B), then A and B are independent. b If P(A|C)P(B|C) and P(A|C)P(B|C), then P(A) P(B).Males and females are observed to react differently to a given set of circumstances. It has been observed that 70% of the females react positively to these circumstances, whereas only 40% of males react positively. A group of 20 people, 15 female and 5 male, was subjected to these circumstances, and the subjects were asked to describe their reactions on a written questionnaire. A response picked at random from the 20 was negative. What is the probability that it was that of a male?A study of Georgia residents suggests that those who worked in shipyards during World War II were subjected to a significantly higher risk of lung cancer (Wall Street Journal, September 21, 1978).3 It was found that approximately 22% of those persons who had lung cancer worked at some prior time in a shipyard. In contrast, only 14% of those who had no lung cancer worked at some prior time in a shipyard. Suppose that the proportion of all Georgians living during World War II who have or will have contracted lung cancer is .04%. Find the percentage of Georgians living during the same period who will contract (or have contracted) lung cancer, given that they have at some prior time worked in a shipyard.131E A plane is missing and is presumed to have equal probability of going down in any of three regions. If a plane is actually down in region i, let 1 i denote the probability that the plane will be found upon a search of the ith region, i = 1, 2, 3. What is the conditional probability that the plane is in a region 1, given that the search of region 1 was unsuccessful? b region 2, given that the search of region 1 was unsuccessful? c region 3, given that the search of region 1 was unsuccessful?A student answers a multiple-choice examination question that offers four possible answers. Suppose the probability that the student knows the answer to the question is .8 and the probability that the student will guess is .2. Assume that if the student guesses, the probability of selecting the correct answer is .25. If the student correctly answers a question, what is the probability that the student really knew the correct answer?Two methods, A and B, are available for teaching a certain industrial skill. The failure rate is 20% for A and 10% for B. However, B is more expensive and hence is used only 30% of the time. (A is used the other 70%.) A worker was taught the skill by one of the methods but failed to learn it correctly. What is the probability that she was taught by method A?Of the travelers arriving at a small airport, 60% fly on major airlines, 30% fly on privately owned planes, and the remainder fly on commercially owned planes not belonging to a major airline. Of those traveling on major airlines, 50% are traveling for business reasons, whereas 60% of those arriving on private planes and 90% of those arriving on other commercially owned planes are traveling for business reasons. Suppose that we randomly select one person arriving at this airport. What is the probability that the person a is traveling on business? b is traveling for business on a privately owned plane? c arrived on a privately owned plane, given that the person is traveling for business reasons? d is traveling on business, given that the person is flying on a commercially owned plane?136E Five identical bowls are labeled 1, 2, 3, 4, and 5. Bowl i contains i white and 5 i black balls, with i = 1, 2, , 5. A bowl is randomly selected and two balls are randomly selected (without replacement) from the contents of the bowl. a What is the probability that both balls selected are white? b Given that both balls selected are white, what is the probability that bowl 3 was selected?138E Refer to Exercise 2.112. Let the random variable Y represent the number of radar sets that detect a particular aircraft. Compute the probabilities associated with each value of Y. 2.112 Three radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each set has a probability of .02 of failing to detect a plane in its area. If an aircraft enters the area, what is the probability that it a goes undetected? b is detected by all three radar sets?Refer to Exercise 2.120. Let the random variable Y represent the number of defective refrigerators found after three refrigerators have been tested. Compute the probabilities for each value of Y. 2.120 Suppose that two defective refrigerators have been included in a shipment of six refrigerators. The buyer begins to test the six refrigerators one at a time. a What is the probability that the last defective refrigerator is found on the fourth test? b What is the probability that no more than four refrigerators need to be tested to locate both of the defective refrigerators? c When given that exactly one of the two defective refrigerators has been located in the first two tests, what is the probability that the remaining defective refrigerator is found in the third or fourth test?Refer again to Exercise 2.120. Let the random variable Y represent the number of the test in which the last defective refrigerator is identified. Compute the probabilities for each value of Y. 2.120 Suppose that two defective refrigerators have been included in a shipment of six refrigerators. The buyer begins to test the six refrigerators one at a time. a What is the probability that the last defective refrigerator is found on the fourth test? b What is the probability that no more than four refrigerators need to be tested to locate both of the defective refrigerators? c When given that exactly one of the two defective refrigerators has been located in the first two tests, what is the probability that the remaining defective refrigerator is found in the third or fourth test?A spinner can land in any of four positions, A, B, C, and D, with equal probability. The spinner is used twice, and the position is noted each time. Let the random variable Y denote the number of positions on which the spinner did not land. Compute the probabilities for each value of Y.180SE 181SE 182SE 183SE A city commissioner claims that 80% of the people living in the city favor garbage collection by contract to a private company over collection by city employees. To test the commissioners claim, 25 city residents are randomly selected, yielding 22 who prefer contracting to a private company. a If the commissioners claim is correct, what is the probability that the sample would contain at least 22 who prefer contracting to a private company? b If the commissioners claim is correct, what is the probability that exactly 22 would prefer contracting to a private company? c Based on observing 22 in a sample of size 25 who prefer contracting to a private company, what do you conclude about the commissioners claim that 80% of city residents prefer contracting to a private company?185SE Refer to Exercises 3.67 and 3.68. Let Y denote the number of the trial on which the first applicant with computer training was found. If each interview costs 30, find the expected value and variance of the total cost incurred interviewing candidates until an applicant with advanced computer training is found. Within what limits would you expect the interview costs to fall? 3.67 Suppose that 30% of the applicants for a certain industrial job possess advanced training in computer programming. Applicants are interviewed sequentially and are selected at random from the pool. Find the probability that the first applicant with advanced training in programming is found on the fifth interview. 3.68 Refer to Exercise 3.67. What is the expected number of applicants who need to be interviewed in order to find the first one with advanced training?Consider the following game: A player throws a fair die repeatedly until he rolls a 2, 3, 4, 5, or 6. In other words, the player continues to throw the die as long as he rolls 1s. When he rolls a non-1, he stops. a What is the probability that the player tosses the die exactly three times? b What is the expected number of rolls needed to obtain the first non-1? c If he rolls a non-1 on the first throw, the player is paid 1. Otherwise, the payoff is doubled for each 1 that the player rolls before rolling a non-1. Thus, the player is paid 2 if he rolls a 1 followed by a non-1; 4 if he rolls two 1s followed by a non-1; 8 if he rolls three 1s followed by a non-1; etc. In general, if we let Y be the number of throws needed to obtain the first non-1, then the player rolls (Y 1) 1s before rolling his first non-1, and he is paid 2Y 1 dollars. What is the expected amount paid to the player?188SE 189SE Toss a balanced die and let Y be the number of dots observed on the upper face. Find the mean and variance of Y. Construct a probability histogram, and locate the interval 2. Verify that Tchebysheffs theorem holds.Two assembly lines I and II have the same rate of defectives in their production of voltage regulators. Five regulators are sampled from each line and tested. Among the total of ten tested regulators, four are defective. Find the probability that exactly two of the defective regulators came from line I.194SE The number of imperfections in the weave of a certain textile has a Poisson distribution with a mean of 4 per square yard. Find the probability that a a 1-square-yard sample will contain at least one imperfection. b 3-square-yard sample will contain at least one imperfection.Refer to Exercise 3.195. The cost of repairing the imperfections in the weave is 10 per imperfection. Find the mean and standard deviation of the repair cost for an 8-square-yard bolt of the textile. 3.195 The number of imperfections in the weave of a certain textile has a Poisson distribution with a mean of 4 per square yard. Find the probability that a a 1-square-yard sample will contain at least one imperfection, b 3-square-yard sample will contain at least one imperfection.The number of bacteria colonies of a certain type in samples of polluted water has a Poisson distribution with a mean of 2 per cubic centimeter (cm3). a If four 1-cm3 samples are independently selected from this water, find the probability that at least one sample will contain one or more bacteria colonies, b How many 1-cm3 samples should be selected in order to have a probability of approximately .95 of seeing at least one bacteria colony?198SE Insulin-dependent diabetes (IDD) is a common chronic disorder in children. The disease occurs most frequently in children of northern European descent, but the incidence ranges from a low of 12 cases per 100,000 per year to a high of more than 40 cases per 100,000 in parts of Finland.4 Let us assume that a region in Europe has an incidence of 30 cases per 100,000 per year and that we randomly select 1000 children from this region. a Can the distribution of the number of cases of IDD among those in the sample be approximated by a Poisson distribution? If so, what is the mean of the approximating Poisson distribution? b What is the probability that we will observe at least two cases of IDD among the 1000 children in the sample?200SE 201SE The number of cars driving past a parking area in a one-minute time interval has a Poisson distribution with mean . The probability that any individual driver actually wants to park his or her car is p. Assume that individuals decide whether to park independently of one another. a If one parking place is available and it will take you one minute to reach the parking area, what is the probability that a space will still be available when you reach the lot? (Assume that no one leaves the lot during the one-minute interval.) b Let W denote the number of drivers who wish to park during a one-minute interval. Derive the probability distribution of W.203SE The probability that any single driver will turn left at an intersection is .2. The left turn lane at this intersection has room for three vehicles. If the left turn lane is empty when the light turns red and five vehicles arrive at this intersection while the light is red, find the probability that the left turn lane will hold the vehicles of all of the drivers who want to turn left.An experiment consists of tossing a fair die until a 6 occurs four times. What is the probability that the process ends after exactly ten tosses with a 6 occurring on the ninth and tenth tosses?Accident records collected by an automobile insurance company give the following information. The probability that an insured driver has an automobile accident is .15. If an accident has occurred, the damage to the vehicle amounts to 20% of its market value with a probability of .80, to 60% of its market value with a probability of .12, and to a total loss with a probability of .08. What premium should the company charge on a 12,000 car so that the expected gain by the company is zero?207SE 208SE 209SE 210SE A merchant stocks a certain perishable item. She knows that on any given day she will have a demand for either two, three, or four of these items with probabilities .1, .4, and .5, respectively. She buys the items for 1.00 each and sells them for 1.20 each. If any are left at the end of the day, they represent a total loss. How many items should the merchant stock in order to maximize her expected daily profit?212SE A lot of N = 100 industrial products contains 40defectives. Let Y be the number of defectives in a random sample of size 20. Find p(10) by using (a) the hypergeometric probability distribution and (b) the binomial probability distribution. Is N large enough that the value for p(10) obtained from the binomial distribution is a good approximation to that obtained using the hypergeometric distribution?For simplicity, let us assume that there are two kinds of drivers. The safe drivers, who are 70% of the population, have probability .1 of causing an accident in a year. The rest of the population are accident makers, who have probability .5 of causing an accident in a year. The insurance premium is 400 times ones probability of causing an accident in the following year. A new subscriber has an accident during the first year. What should be his insurance premium for the next year?216SE 217SE 218SE When the health department tested private wells in a county for two impurities commonly found in drinking water, it found that 20% of the wells had neither impurity, 40% had impurity A, and 50% had impurity B. (Obviously, some had both impurities.) If a well is randomly chosen from those in the county, find the probability distribution for Y, the number of impurities found in the well.You and a friend play a game where you each toss a balanced coin. If the upper faces on the coins are both tails, you win 1; if the faces are both heads, you win 2; if the coins do not match (one shows a head, the other a tail), you lose 1 (win (1)). Give the probability distribution for your winnings. Y, on a single play of this game.A group of four components is known to contain two defectives. An inspector tests the components one at a time until the two defectives are located. Once she locates the two defectives, she stops testing, but the second defective is tested to ensure accuracy. Let Y denote the number of the test on which the second defective is found. Find the probability distribution for Y.Consider a system of water flowing through valves from A to B. (See the accompanying diagram.) Valves 1, 2, and 3 operate independently, and each correctly opens on signal with probability .8. Find the probability distribution for Y. the number of open paths from A to B after the signal is given. (Note that Y can take on the values 0, 1, and 2.)A problem in a test given to small children asks them to match each of three pictures of animals to the word identifying that animal. If a child assigns the three words at random to the three pictures, find the probability distribution for Y, the number of correct matches.Five balls, numbered 1, 2, 3, 4, and 5, are placed in an urn. Two balls are randomly selected from the five, and their numbers noted. Find the probability distribution for the following: a. The largest of the two sampled numbers b. The sum of the two sampled numbers Each of three balls are randomly placed into one of three bowls. Find the probability distribution for Y = the number of empty bowls.A single cell can either die, with probability .1, or split into two cells, with probability .9, producing a new generation of cells. Each cell in the new generation dies or splits into two cells independently with the same probabilities as the initial cell. Find the probability distribution for the number of cells in the next generation.In order to verify the accuracy of their financial accounts, companies use auditors on a regular basis to verify accounting entries. The companys employees make erroneous entries 5% of the time. Suppose that an auditor randomly checks three entries. a. Find the probability distribution for Y, the number of errors detected by the auditor, b. Construct a probability histogram for p(y). c. Find the probability that the auditor will detect more than one error.A rental agency, which leases heavy equipment by the day, has found that one expensive piece of equipment is leased, on the average, only one day in five. If rental on one day is independent of rental on any other day, find the probability distribution of Y, the number of days between a pair of rentals.Persons entering a blood bank are such that 1 in 3 have type O+ blood and 1 in 15 have type Oblood. Consider three randomly selected donors for the blood bank. Let X denote the number of donors with type O+ blood and Y denote the number with type O blood. Find the probability distributions for X and Y. Also find the probability distribution for X + Y, the number of donors who have type O blood.Let Y be a random variable with p(y) given in the accompanying table. Find E(Y), E(1/Y), E(Y2 1), and V(Y).Refer to the coin-tossing game in Exercise 3.2. Calculate the mean and variance of Y, your winnings on a single play of the game. Note that E(Y) 0. How much should you pay to play this game if your net winnings, the difference between the payoff and cost of playing, are to have mean 0? 3.2 You and a friend play a game where you each toss a balanced coin. If the upper faces on the coins are both tails, you win 1; if the faces are both heads, you win 2; if the coins do not match (one shows a head, the other a tail), you lose 1 (win (1)). Give the probability distribution for your winnings, Y, on a single play of this game.The maximum patent life for a new drug is 17 years. Subtracting the length of time required by the FDA for testing and approval of the drug provides the actual patent life for the drugthat is, the length of time that the company has to recover research and development costs and to make a profit. The distribution of the lengths of actual patent lives for new drugs is given below: a. Find the mean patent life for a new drug. b. Find the standard deviation of Y = the length of life of a randomly selected new drug. c. What is the probability that the value of Y falls in the interval 2?Who is the king of late night TV? An Internet survey estimates that, when given a choice between David Letterman and Jay Leno, 52% of the population prefers to watch Jay Leno. Three late night TV watchers are randomly selected and asked which of the two talk show hosts they prefer. a. Find the probability distribution for Y, the number of viewers in the sample who prefer Leno. b. Construct a probability histogram for p(y). c. What is the probability that exactly one of the three viewers prefers Leno? d. What are the mean and standard deviation for Y? e. What is the probability that the number of viewers favoring Leno falls within 2 standard deviations of the mean?16E Refer to Exercise 3.7. Find the mean and standard deviation for Y = the number of empty bowls. What is the probability that the value of Y falls within 2 standard deviations of the mean? 3.7 Each of three balls are randomly placed into one of three bowls. Find the probability distribution for Y = the number of empty bowls.Refer to Exercise 3.8. What is the mean number of cells in the second generation? 3.8 A single cell can either die, with probability .1, or split into two cells, with probability .9, producing a new generation of cells. Each cell in the new generation dies or splits into two cells independently with the same probabilities as the initial cell. Find the probability distribution for the number of cells in the next generation.An insurance company issues a one-year 1000 policy insuring against an occurrence A that historically happens to 2 out of every 100 owners of the policy. Administrative fees are 15 per policy and are not part of the companys profit. How much should the company charge for the policy if it requires that the expected profit per policy be 50? [Hint: If C is the premium for the policy, the companys profit is C 15 if A does not occur and C 15 1000 if A does occur.]A manufacturing company ships its product in two different sizes of truck trailers. Each shipment is made in a trailer with dimensions 8 feet 10 feet 30 feet or 8 feet 10 feet 40 feet. If 30% of its shipments are made by using 30-foot trailers and 70% by using 40-foot trailers, find the mean volume shipped per trailer load. (Assume that the trailers are always full.)The number N of residential homes that a fire company can serve depends on the distance r (in city blocks) that a fire engine can cover in a specified (fixed) period of time. If we assume that N is proportional to the area of a circle R blocks from the firehouse, then N = C R2, where C is a constant, = 3.1416 , and R, a random variable, is the number of blocks that a fire engine can move in the specified time interval. For a particular fire company, C = 8, the probability distribution for R is as shown in the accompanying table, and p(r) = 0 for r 20 and r 27.A single fair die is tossed once. Let Y be the number facing up. Find the expected value and variance of Y.In a gambling game a person draws a single card from an ordinary 52-card playing deck. A person is paid 15 for drawing a jack or a queen and 5 for drawing a king or an ace. A person who draws any other card pays 4. If a person plays this game, what is the expected gain?Approximately 10% of the glass bottles coming off a production line have serious flaws in the glass. If two bottles arc randomly selected, find the mean and variance of the number of bottles that have serious flaws.Two construction contracts are to be randomly assigned to one or more of three firms: I, II, and III. Any firm may receive both contracts. If each contract will yield a profit of 90,000 for the firm, find the expected profit for firm I. If firms I and II are actually owned by the same individual, what is the owners expected total profit?A heavy-equipment salesperson can contact either one or two customers per day with probability 1/3 and 2/3, respectively. Each contact will result in either no sale or a 50,000 sale, with the probabilities .9 and .1, respectively. Give the probability distribution for daily sales. Find the mean and standard deviation of the daily sales.A potential customer for an 85,000 fire insurance policy possesses a home in an area that, according to experience, may sustain a total loss in a given year with probability of .001 and a 50% loss with probability .01. Ignoring all other partial losses, what premium should the insurance company charge for a yearly policy in order to break even on all 85,000 policies in this area?Refer to Exercise 3.3. If the cost of testing a component is 2 and the cost of repairing a defective is 4, find the expected total cost for testing and repairing the lot. 3.3 A group of four components is known to contain two defectives. An inspector tests the components one at a time until the two defectives are located. Once she locates the two defectives, she stops testing, but the second defective is tested to ensure accuracy. Let Y denote the number of the test on which the second defective is found. Find the probability distribution for Y.If Y is a discrete random variable that assigns positive probabilities to only the positive integers, show that E(Y)=i=1P(Yk).Suppose that Y is a discrete random variable with mean and variance 2 and let X = Y + 1. a. Do you expect the mean of X to be larger than, smaller than, or equal to = E(Y)? Why? b. Use Theorems 3.3 and 3.5 to express E(X) = E(Y + 1) in terms of = E(Y). Does this result agree with your answer to part (a)? c. Recalling that the variance is a measure of spread or dispersion, do you expect the variance of X to be larger than, smaller than, or equal to 2 = V(Y)? Why? d. Use Definition 3.5 and the result in part (b) to show that V(X)=E{[(XE(X)]2}=E[(Y)2]=2; that is, X = Y + 1 and Y have equal variances.Suppose that Y is a discrete random variable with mean and variance 2 and let W = 2Y. a. Do you expect the mean of W to be larger than, smaller than, or equal to = E(Y)? Why? b. Use Theorem 3.4 to express E(W) = E(2Y) in terms of = E(Y). Does this result agree with your answer to part (a)? c. Recalling that the variance is a measure of spread or dispersion, do you expect the variance of W to be larger than, smaller than, or equal to 2 = V(Y)? Why? d. Use Definition 3.5 and the result in part (b) to show that V(W)=E{[WE(W)]2}=E[4(Y)2]=42; that is, W = 2Y has variance four times that of Y.Suppose that Y is a discrete random variable with mean and variance 2 and let U = Y/10. a. Do you expect the mean of U to be larger than, smaller than, or equal to = E(Y)? Why? b. Use Theorem 3.4 to express E(U) = E(Y/10) in terms of = E(Y). Does this result agree with your answer to part (a)? c. Recalling that the variance is a measure of spread or dispersion, do you expect the variance of U to be larger than, smaller than, or equal to 2 = V(Y)? Why? d. Use Definition 3.5 and the result in part (b) to show that V(U)=E{[UE(U)]2}=E[0.1(Y)2]=0.12; that is, U = Y/10 has variance .01 times that of Y.Let Y be a discrete random variable with mean and variance 2. If a and b are constants, use Theorems 3.3 through 3.6 to prove that a. E(aY+b)=aE(Y)+b=a+b. b. V(aY+b)=a2V(Y)a22.The manager of a stockroom in a factory has constructed the following probability distribution for the daily demand (number of times used) for a particular tool. It costs the factory 10 each time the tool is used. Find the mean and variance of the daily cost for use of the tool.Consider the population of voters described in Example 3.6. Suppose that there are N = 5000 voters in the population, 40% of whom favor Jones. Identify the event favors Jones as a success S. It is evident that the probability of S on trial 1 is .40. Consider the event B that S occurs on the second trial. Then B can occur two ways: The first two trials are both successes or the first trial is a failure and the second is a success. Show that P(B) = .4. What is P(B| the first trial is S)? Does this conditional probability differ markedly from P(B)?a. A meteorologist in Denver recorded Y = the number of days of rain during a 30-day period. Does Y have a binomial distribution? If so, are the values of both n and p given? b. A market research firm has hired operators who conduct telephone surveys. A computer is used to randomly dial a telephone number, and the operator asks the answering person whether she has time to answer some questions. Let Y = the number of calls made until the first person replies that she is willing to answer the questions. Is this a binomial experiment? Explain.In 2003, the average combined SAT score (math and verbal) for college-bound students in the United States was 1026. Suppose that approximately 45% of all high school graduates took this test and that 100 high school graduates are randomly selected from among all high school grads in the United States. Which of the following random variables has a distribution that can be approximated by a binomial distribution? Whenever possible, give the values for n and p. a. The number of students who took the SAT b. The scores of the 100 students in the sample c. The number of students in the sample who scored above average on the SAT d. The amount of time required by each student to complete the SAT e. The number of female high school grads in the sample The manufacturer of a low-calorie dairy drink wishes to compare the taste appeal of a new formula (formula B) with that of the standard formula (formula A). Each of four judges is given three glasses in random order, two containing formula A and the other containing formula B. Each judge is asked to state which glass he or she most enjoyed. Suppose that the two formulas are equally attractive. Let Y be the number of judges stating a preference for the new formula. a. Find the probability function for Y. b. What is the probability that at least three of the four judges state a preference for the new formula? c. Find the expected value of Y. d. Find the variance of Y.A complex electronic system is built with a certain number of backup components in its subsystems. One subsystem has four identical components, each with a probability of .2 of failing in less than 1000 hours. The subsystem will operate if any two of the four components are operating. Assume that the components operate independently. Find the probability that a. exactly two of the four components last longer than 1000 hours, b. the subsystem operates longer than 1000 hours.The probability that a patient recovers from a stomach disease is .8. Suppose 20 people are known to have contracted this disease. What is the probability that a. exactly 14 recover? b. at least 10 recover? c. at least 14 but not more than 18 recover? d. at most 16 recover?A multiple-choice examination has 15 questions, each with five possible answers, only one of which is correct. Suppose that one of the students who takes the examination answers each of the questions with an independent random guess. What is the probability that he answers at least ten questions correctly?Refer to Exercise 3.41. What is the probability that a student answers at least ten questions correctly if a. for each question, the student can correctly eliminate one of the wrong answers and subsequently answers each of the questions with an independent random guess among the remaining answers? b. he can correctly eliminate two wrong answers for each question and randomly chooses from among the remaining answers? 3.41 A multiple-choice examination has 15 questions, each with five possible answers, only one of which is correct. Suppose that one of the students who takes the examination answers each of the questions with an independent random guess. What is the probability that he answers at least ten questions correctly?Many utility companies promote energy conservation by offering discount rates to consumers who keep their energy usage below certain established subsidy standards. A recent EPA report notes that 70% of the island residents of Puerto Rico have reduced their electricity usage sufficiently to qualify for discounted rates. If five residential subscribers are randomly selected from San Juan, Puerto Rico, find the probability of each of the following events: a. All five qualify for the favorable rates, b. At least four qualify for the favorable rates.44E A fire-detection device utilizes three temperature-sensitive cells acting independently of each other in such a manner that any one or more may activate the alarm. Each cell possesses a probability of p = .8 of activating the alarm when the temperature reaches 100 Celsius or more. Let Y equal the number of cells activating the alarm when the temperature reaches 100. a. Find the probability distribution for Y. b. Find the probability that the alarm will function when the temperature reaches 100.

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