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All Textbook Solutions for Mathematical Applications for the Management, Life, and Social Sciences
37. Education A mathematics class consists of
16 engineering majors, 12 science majors, and 4 liberal arts majors.
What is the probability that a student selected at random will be a science or liberal arts major?
What is the probability that a student selected at random will be an engineering or science major?
Five of the engineering students, 6 of the science majors, and 2 of the liberal arts majors are female. What is the probability that a student selected at random is an engineering major or a female?
38. Parts delivery Repairing a copy machine requires that two parts be delivered from two suppliers. The probability that part A will be delivered on Thursday is 0.6, and the probability that part B will be delivered on Thursday is 0.8. If the probability that one or the other part will arrive on Thursday is 0.9, what is the probability that both will be delivered on Thursday?
Oil drilling The table summarizes the opinions of various groups on the issue of oil drilling in national parks. Use this table to calculate probabilities in Problems 39-42.
Opinion Whites Nonwhites Total
Reps. Dems, Reps. Dems.
Favor 300 100 25 10 435
Oppose 100 250 25 190 565
Total 400 350 50 200 1000
39. Find the probability that an individual chosen at
random is a Republican or favors oil drilling in national parks.
40E41E42E43. Drinking age A survey questioned 1000 people regarding lowering the legal drinking age from 21 to 19. Of the 560 who favored lowering the age, 390 were male. Of the 440 opposition responses, 160 were male. A person is selected at random.
What is the probability that the person is a male or favors lowering the age?
What is the probability that the person is a female or favors lowering the age?
What is the probability that the person is a female or opposes lowering the age?
44. Job bids Three construction companies have bid for a job. Max knows that the two companies with which he is competing have probabilities 1/3 and 1/6, respectively, of getting the job. What is the probability that Max will get the job?
45E46E47E48ECHECKPOINT
1. Suppose that one ball is drawn from a bag containing 4 red balls numbered 1, 2, 3, 4 and 3 white balls numbered 5, 6, 7. What is the probability that the ball is white, given that it is an even-numbered ball?
2CPCHECKPOINT
3. Two balls are drawn with replacement from an urn that contains 3 red and 4 green halls.
(a) Are these draws independent? Explain.
(b) Find the probability of getting a green ball on both draws.
1. A card is drawn from a deck of 52 playing cards. Given that it is a red card, what is the probability that
(a) it is a heart? (b) it is a king
2E3. A die has been “loaded so that the probability of rolling any even number is 2/9 and the probability of rolling any odd number is 1/9. What is the probability of
(a) rolling a 6, given that an even number is rolled?
(b) rolling a 3, given that a number divisible by 3 is rolled?
4. A die has been “loaded” so that the probability of rolling any even number is 2/9 and the probability of rolling any odd number is 1/9. What is the probability of
(a) rolling a 5?
(b) rolling a 5, given that an even number is rolled?
(c) rolling a 5, given that an odd number is rolled?
5. A bag contains 9 red balls numbered 1, 2, 3, 4, 5, 6, 7, 8, 9 and 6 white balls numbered 10, 11, 12, 13, 14, 15. One ball is drawn from the bag. What is the probability that the ball is red, given that the ball is even-numbered?
6E7. A bag contains 4 red balls and 6 white balls. Two balls are drawn without replacement.
(a) What is the probability that the second ball is white, given that the first ball is red?
(b) What is the probability that the second ball is red, given that the first ball is white?
(c) Answer part (a) if the first ball is replaced before the second is drawn.
8. A fair die is rolled. Find the probability that the result is a 4, given that the result is even.
9. A fair coin is tossed 3 times. Find the probability of
(a) throwing 3 heads, given that the first toss is a head.
(b) throwing 3 heads, given that the first two tosses result in heads.
10. A fair coin is tossed 14 times. What is the probability of tossing 14 heads, given that the first 13 tosses are heads?
11. A die is thrown twice. What is the probability that a 3 will result the first time and a 6 the second time?
12E13E14E15. (a) A box contains 3 red balls, 2 white balls, and 5 black balls. Two balls are drawn at random from the box (with replacement of the first before the second is drawn). What is the probability of getting a red ball on the first draw and a white ball on the second?
(b) Answer the question in part (a) if the first ball is not replaced before the second ball is drawn.
(c) Are the events in part (a) or in part (b) independent? Explain.
16E17. Two balls are drawn from a bag containing 3 white balls and 2 red balls. If the first ball is replaced before the second is drawn, what is the probability that (a) both balls are red? (b) both balls are white?
(c) the first ball is red and the second ball is white?
(d) one of the balls is black?
18E19. A bag contains 9 nickels, 4 dimes, and 5 quarters. If you draw 3 coins at random from the bag without replacement, what is the probability that you will get a nickel, a quarter, and a nickel in that order?
20E21. A red ball and 4 white balls are in a box. If two balls are drawn without replacement, what is the probability
(a) of getting a red ball on the first draw and a white ball on the second?
(b) of getting 2 white balls?
(c) of getting 2 red balls?
22. From a deck of 52 playing cards, two cards are drawn, one after the other without replacement. What is the probability that
(a) the first will be a king and the second will be a jack?
(b) the first will be a king and the second will be a jack of the same suit?
23. One card is drawn at random from a deck of 52 cards. The first card is not replaced, and a second card is drawn. Find the probability that
(a) both cards are spades.
(b) the first card is a heart and the second is a club.
24E25. Two balls are drawn without replacement from a bag containing 13 red balls numbered 1-13 and 5 white balls numbered 14-18. What is the probability that
(a) the second ball is red, given that the first ball is white?
(b) both balls are even-numbered?
(c) the first ball is red and even-numbered and the second ball is even-numbered?
26E27E28ECarbon emissions The table gives the results of a survey of 1000 people about reducing industries’ carbon emissions. Use the table to answer Problems 27-30.
Favor Oppose No Opinion Total
Democrat 310 150 60 520
Republican 125 345 10 480
Total 435 495 70 1000
29. If the group surveyed represents the people of the United States, what is the probability that a citizen selected at random will favor reducing industries’ carbon emissions, given that the person is a Republican?
30EOil drilling Suppose the table summarizes the opinions of various groups on the issue of oil drilling in national parks. Use this table to calculate the empirical probabilities in Problems 31-38.
Opinion Whites Non-Whites Total
Reps. Dems. Reps. Dems.
Favor 300 100 25 10 435
Oppose 100 250 25 190 565
Total 400 350 50 200 1000
31. Given that a randomly selected individual is non-White, find the probability that he or she opposes oil drilling in national parks.
Oil drilling Suppose the table summarizes the opinions of various groups on the issue of oil drilling in national parks. Use this table to calculate the empirical probabilities in Problems 31-38.
Opinion Whites Non-Whites Total
Reps. Dems. Reps. Dems.
Favor 300 100 25 10 435
Oppose 100 250 25 190 565
Total 400 350 50 200 1000
32. Given that a randomly selected individual is a Democrat, find the probability that he or she opposes oil drilling in national parks.
Oil drilling Suppose the table summarizes the opinions of various groups on the issue of oil drilling in national parks. Use this table to calculate the empirical probabilities in Problems 31-38.
Opinion Whites Non-Whites Total
Reps. Dems. Reps. Dems.
Favor 300 100 25 10 435
Oppose 100 250 25 190 565
Total 400 350 50 200 1000
33. Given that a randomly selected individual is in favor of oil drilling in national parks, find the probability that he or she is a Republican.
Oil drilling Suppose the table summarizes the opinions of various groups on the issue of oil drilling in national parks. Use this table to calculate the empirical probabilities in Problems 31-38.
Opinion Whites Non-Whites Total
Reps. Dems. Reps. Dems.
Favor 300 100 25 10 435
Oppose 100 250 25 190 565
Total 400 350 50 200 1000
34. Given that a randomly selected individual is opposed to oil drilling in national parks, find the probability that he or she is a Democrat.
Oil drilling Suppose the table summarizes the opinions of various groups on the issue of oil drilling in national parks. Use this table to calculate the empirical probabilities in Problems 31-38.
Opinion Whites Non-Whites Total
Reps. Dems. Reps. Dems.
Favor 300 100 25 10 435
Oppose 100 250 25 190 565
Total 400 350 50 200 1000
35. Find the probability that a person who favors oil drilling in national parks is non-White.
Oil drilling Suppose the table summarizes the opinions of various groups on the issue of oil drilling in national parks. Use this table to calculate the empirical probabilities in Problems 31-38.
Whites Non-Whites
Opinion Reps. Dems. Reps. Dems. Total
Favor 300 100 25 10 435
Oppose 100 250 25 190 565
Total 400 350 50 200 1000
36. Find the probability that an individual is White and opposes oil drilling in national parks.
Oil drilling Suppose the table summarizes the opinions of various groups on the issue of oil drilling in national parks. Use this table to calculate the empirical probabilities in Problems 31-38.
Whites Non-Whites
Opinion Reps. Dems. Reps. Dems. Total
Favor 300 100 25 10 435
Oppose 100 250 25 190 565
Total 400 350 50 200 1000
37. Find the probability that an individual is a White Republican opposed to oil drilling in national parks.
Oil drilling Suppose the table summarizes the opinions of various groups on the issue of oil drilling in national parks. Use this table to calculate the empirical probabilities in Problems 31-38.
Whites Non-Whites
Opinion Reps. Dems. Reps. Dems. Total
Favor 300 100 25 10 435
Oppose 100 250 25 190 565
Total 400 350 50 200 1000
38. Find the probability that an individual is a Democrat and opposes oil drilling in national parks.
39E40E41E42E43. Quality control Each computer component that the Peggos Company produces is independently tested twice before it is shipped. There is a 0.7 probability that a defective component will be so identified by the first test and a 0.8 probability that it will be identified as being defective by the second test. What is the probability that a defective component will not be identified as defective before it is shipped?
44E45. Lactose intolerance Lactose intolerance affects about 50% of Hispanic Americans, and 9% of the residents of the United States are Hispanic (Source: Jean Carper, “Eat Smart,” USA Weekend). If a U.S. resident is selected at random, what is the probability that the person will be Hispanic and have lactose intolerance?
46. Quality control If 3% of all light bulbs a company manufactures are defective, the probability of any one bulb being defective is 0.03. What is the probability that three bulbs drawn independently from the company’s stock will be defective?
47. Quality control To test its shotgun shells, a company fires 5 of them. What is the probability that all 5 will fire properly if 5% of the company’s shells are actually defective?
48E49E50E51E52E53. Maintenance Twenty-three percent of the cars owned by a car rental firm have some defect. What is the probability that of 3 cars selected at random,
(a) none has a defect?
(b) at least one has a defect?
54E55. Testing An unprepared student must take a 7-question multiple-choice test that has 5 possible answers per question. If the student can eliminate two of the possible answers on the first three questions, and if she guesses on every question, what is the probability that
(a) she will answer every question correctly?
(b) she will answer every question incorrectly?
(c) she will answer at least one question correctly?
56E57E58. Racing Suppose that the odds that Blackjack will win a race are 1 to 3 and the odds that Snowball will win the same race are 1 to 5. If only one horse can win, what odds should be given that one of these two horses will win?
59E60E61E62. Crime In an actual case.’ probability was used to convict a couple of mugging an elderly woman. Shortly after the mugging, a young White woman with blonde hair worn in a ponytail was seen running from the scene of the crime and entering a yellow car that was driven by a Black man with a beard. A couple matching this description was arrested for the crime. A prosecuting attorney argued that the couple arrested had to be the couple at the scene of the crime because the probability of a second couple matching the description was very small. He estimated the probabilities of six events as follows:
Probability of Black-While couple: 1/1000
Probability of Black man: 1/3
Probability of bearded man: 1/10
Probability of blonde woman: ¼
Probability of hair in ponytail: 1/10
Probability of yellow car: 1/10
He multiplied these probabilities and concluded that the probability that another couple would have these characteristics is 1/12,000,000. On the basis of this circumstantial evidence, the couple was convicted and sent to prison. The conviction was overturned by the state supreme court because the prosecutor made an incorrect assumption. What error do you think he made?
’Time. January 8.1965, p. 42. and April 26,1968, p. 41.
CHECKPOINT
1. Urn I contains 3 gold coins, urn II contains I gold coin and 2 silver coins, and urn III contains I gold coin and I silver coin. If an urn is selected at random and a coin is drawn from the urn, construct a probability tree and find the probability that a gold coin will be drawn.
2CPIn Problems 1-12, use probability trees to find the probabilities of the indicated outcomes.
1. A bag contains 5 coins, 4 of which are fair and 1 that has a head on each side. If a coin is selected from the bag and tossed twice, what is the probability of obtaining
(a) 2 heads? (b) 2 tails?
2EIn Problems 1-12, use probability trees to find the probabilities of the indicated outcomes.
3. An urn contains 4 red, 5 white, and 6 black balls. One ball is drawn from the urn, it is not replaced, and a second ball is drawn.
(a) What is the probability that both balls are white?
(b) What is the probability that one ball is white and one is red?
(c) What is the probability that at least one ball is black?
4EIn Problems 1-12, use probability trees to find the probabilities of the indicated outcomes.
5. Three balls are drawn without replacement from a bag containing 4 red balls and 5 white balls. Find the probability that
(a) three white balls are drawn.
(b) two white balls and one red ball are drawn.
(c) the third ball drawn is red.
6EIn Problems 1-12, use probability trees to find the probabilities of the indicated outcomes.
7. A bag contains 4 white balls and 6 red balls. Three balls are drawn without replacement from the bag.
(a) What is the probability that all three balls are white?
(b) What is the probability that exactly one ball is white?
(c) What is the probability that at least one ball is white?
In Problems 1-12, use probability trees to find the probabilities of the indicated outcomes.
8. A bag contains 5 coins, of which 4 are fair; the remaining coin has a head on both sides. If a coin is selected at random from the bag and tossed three times, what is the probability that
(a) heads will occur exactly twice?
(b) heads will occur at least twice?
9E10EIn Problems 1-12, use probability trees to find the probabilities of the indicated outcomes.
11. Two chips are drawn without replacement from a bag that contains 9 black chips numbered 1-9 and 4 white chips numbered 10-13. Find the probability that
(a) the first chip is odd-numbered given that the second is even-numbered.
(b) the first chip is white and the second is black.
(c) the first chip is even-numbered or the second chip is odd-numbered.
(d) the second chip is white given that the first chip is even-numbered.
In Problems 1-12, use probability trees to find the probabilities of the indicated outcomes.
12. Two balls are drawn without replacement from a bag containing 13 red balls numbered 1-13 and 5 white balls numbered 14-18. What is the probability that
(a) the second ball is even-numbered, given that the first ball is even-numbered?
(b) the first ball is red and the second ball is even-numbered?
(c) the first ball is even-numbered and the second is white?
(d) the first ball is even-numbered given that the second is white?
In Problems 13-16, use (a) a probability tree and (b) Bayes’ formula to find the probabilities.
In Problems 13 and 14, each of urns I and II has 5 red balls, 3 white balls, and 2 green balls. Urn III has 1 red ball, 1 white ball, and 8 green balls.
13. An urn is selected at random, and a ball is drawn. If the ball is green, find the probability that urn III was selected.
14EIn Problems 13-16, use (a) a probability tree and (b) Bayes’ formula to find the probabilities.
In Problems 13 and 14, each of urns I and II has 5 red balls, 3 white balls, and 2 green balls. Urn III has 1 red ball, 1 white ball, and 8 green balls.
15. Three urns contain coins. Urn I contains 3 gold coins, urn II contains 1 gold coin and 1 silver coin, and urn III contains 2 silver coins. An urn is selected at random, and a coin is drawn from the urn. If the selected coin is gold, what is the probability that urn 1 was selected?
In Problems 13-16, use (a) a probability tree and (b) Bayes’ formula to find the probabilities.
In Problems 13 and 14, each of urns I and II has 5 red balls, 3 white balls, and 2 green balls. Urn III has 1 red ball, 1 white ball, and 8 green balls.
16. In Problem 15, what is the probability that urn III was selected if the coin selected was silver?
17. Lactose intolerance Lactose intolerance affects about of non-Hispanic White Americans; of Hispanic Americans; and of African, Asian, and Native Americans (Source: Jean Carper, “Eat Smart,” USA Weekend). Seventy-six percent of U.S. residents are non-Hispanic Whites; of them are Hispanic; and are African, Asian, or Native American. If a person is selected from this group of people, what is the probability that the person will have lactose intolerance?
18. Genetics What is the probability that a couple will have at least two sons if they plan to have 3 children and if the probability of having a son equals the probability of having a daughter?
19. Marksmanship Suppose that a marksman hits the bull’s-eye 15,000 times in 50,000 shots. If the next 4 shots are independent, find the probability that
(a) the next 4 shots hit the bull’s-eye.
(b) two of the next 4 shots hit the bull’s-eye.
20. Majors In a random survey of students concerning student activities, 30 engineering majors, 25 business majors, 20 science majors, and 15 liberal arts majors were selected. If two students are selected at random, what is the probability of getting
(a) two science majors?
(b) a science major and an engineering major?
21. Quality control Suppose a box contains 3 defective batteries and 12 good batteries. If 2 batteries are drawn from the box without replacement, what is the probability that
(a) the first one is good and the second one is defective?
(b) the first one is defective and the second one is good?
(c) one of the batteries drawn is good and one of them is defective?
22. Education The probability that an individual without a college education earns more than $100,000 is 0.2, whereas the probability that a person with a B.S. or higher degree earns more than $100,000 is 0.6. The probability that a person chosen at random has a B.S. degree is 0.3. What is the probability that a person has at least a B.S. degree if it is known that be or she earns more than $100,000?
23. Alcoholism A small town has 8000 adult males and 6000 adult females. A sociologist conducted a survey and found that 40% of the males and 30% of the females
drink heavily. An adult is selected at random from the town.
(a) What is the probability the person is a male?
(b) What is the probability the person drinks heavily?
(c) What is the probability the person is a male or drinks heavily?
(d) What is the probability the person is a male if it is known that the person drinks heavily?
24E25E26. Insurance rates Ideally, auto insurance rates are lower for good drivers than for bad drivers, but an insurer needs to be able to tell which type of driver a client is. Assume that all drivers are considered as either “good drivers” or “bad drivers” and that the probability of a random driver being a “good driver” is 0.80. In addition,
suppose that the probability of 2 accidents in a year is 0.02 for good drivers and 0.10 for bad drivers.
(a) If an insurer has no additional information about an applicant’s driving history, what is the probability that the individual is a bad driver?
(b) If the insurer knows that the applicant had 2 accidents in the past year, what is the probability that he or she is a bad driver?
27. Pregnancy test A self-administered pregnancy test detects 85% of those who are pregnant but does not detect pregnancy in 15%. It is 90% accurate in indicating women who are not pregnant but indicates 10% of this group as being pregnant. Suppose it is known that 1% of the women in a neighborhood are pregnant. If a woman is chosen at random from those living in this neighborhood and if the test indicates she is pregnant, what is the probability that she really is?
28E29. Survey In a Student Activity Participation Study (SAPS) survey, 30 engineering majors, 25 business majors, 20 science majors, 15 liberal arts majors, and 10 human development majors were selected. Ten of the engineering, 12 of the science, 6 of the human development, 13 of the business, and 8 of the liberal arts majors selected for the study were female.
(a) What is the probability of selecting a female if one person from this group is selected randomly?
(b) What is the probability that a student selected randomly from this group is a science major, given that she is female?
30ECHECKPOINT
1. If a state permits either a letter or a nonzero digit to be used in each of six places on its license plates, how many different plates can it issue?
2CP3CP4CP5CP1. Compute .
2E3. Compute .
4. Compute .
5. Compute .
6E7. How many four-digit numbers can be formed from the digits 1, 3, 5, 7, 8, and 9
(a) if each digit may be used once in each number?
(b) if each digit may be used repeatedly in each number?
8E9. Compute .
10E11E12E13E14E15. Compute .
16. Compute .
17. Compute .
18E19. Compute .
20E21E22E23. Binz, not Benz Mercedes Benz E-Class sedans are converted to limousines by a coach company. The limousines, named Binz, come in two models, XL Six Door and XL Vis-à-Vis, and are available with a choice of 2 gasoline engines or 2 diesel engines.
(a) If no other options are considered, how many different Binz limos are available?
(b) If, in addition, 5 option packages and 6 colors are available, how many different Binz limos are available?
24. Ice cream cones Baskin-Robbins’logo includes “31” because its stores offer 31 flavors of ice cream cones in sizes small, medium, and large.
(a) How many different selections of cones are possible if each cone has one flavor?
(b) If 4 toppings are available to put on a cone and a cone can be bought without a topping, how many different selections of ice cream cones are available if each cone has one flavor?
25E26. Menu selections A restaurant has an early dining fixe prix (fixed price) dinner special that offers one soup or salad from 5 available, one entree from 5 chef’s specials, and one dessert from 4 choices.
(a) How many different fixe prix dinners are possible?
(b) Suppose one particular diner wants a salad and a fish entree with any of the dessert choices.
If 3 different salads and 2 fish entrees are available, how many fewer meal choices does this diner have?
27E28. Racing Eight horses are entered in a race. In how many ways can the horses finish? Assume no ties.
29E30. Testing An examination consists of 12 questions. If 10 questions must be answered, find the number of different orders in which a student can answer the questions.
31E32E33. Management A department store manager wants to display 6 brands of a product along one shelf of an aisle. In how many ways can he arrange the brands?
34. Call letters The call letters for radio stations begin with K or W, followed by 3 additional letters. How many sets of call letters having 4 letters are possible?
35E36. Testing How many ways can a 10-question multiple- choice test be answered if each question has 4 possible answers?
37E38E39E40. Astronauts If 8 people are qualified for the next flight of a new space vehicle, how many different groups of 3 people can be chosen for the flight?
41. Sales A traveling salesperson has 30 products to sell hut has room in her sample case for only 20 of the products. If all the products are the same size, in how many ways can she select 20 different products for her case?
42E43E44E45E46E47. Committees In how many ways can a committee consisting of 6 men and 6 women be selected from a group consisting of 20 men and 22 women?
48E49E50. Auto dealerships An auto dealer has 4 cars, 2 trucks, and 3 SUVs that it plans to display in a line across the front of the dealership.
In how many orders can these vehicles be displayed?
If the cars are grouped together at the left end of the line, in how many orders can the vehicles be displayed?
CHECKPOINT
1. If three wires (red, black, and white) are randomly attached to a three-way switch (which has 3 poles to which wires can be attached), what is the probability that the wires will be attached at random in the one order that makes the switch work properly?
2CP1. Education If a child is given cards with A, C, D, G, O, and T on them, what is the probability that he or she could spell DOG by guessing the correct arrangement of 3 cards from the 6?
2EPolitics A poll asks voters to rank Social Security, the economy, the war on terror, health care, and education in order of importance.
How many rankings are possible?
What is the probability that one reply chosen at random has the issues ranked in the order they appear on the survey?
4. Photography Two men and a woman are lined up to have their picture taken. If they are arranged at random, what is the probability that
the woman will be on the left in the picture?
the woman will be in the middle in the picture?
5. License plates Suppose that all license plates in a state have three letters then three digits. If a plate is chosen at random, what is the probability that all three letters and all three numbers on the plate will be different?
6E7. ATMs
(a) An automatic teller machine requires that each customer enter a four-digit personal identification number (PIN) when he or she inserts a bank card. If a person finds a bank card and guesses at a PIN to use the card fraudulently, what is the probability that the person will succeed in one attempt?
(b) If the person knows that the PIN will not have any digit repeated, what is the probability that the person will succeed in guessing in one attempt?
8. Keys Keys for antique General Motors cars had six parts with three patterns for each part.
(a) How many different key designs are possible for these cars?
(b) If you find an antique GM key and own an antique GM car, what is the probability that it will fit your trunk?
9. Phone numbers If the first digit of a seven-digit phone number cannot be a 0 or a 1, what is the probability that a number chosen at random will have all seven digits the same?
10E11. Rewards As a reward for a record year, the Ace Software Company is randomly selecting 4 people from its 500 employees for a free trip to Hawaii, but it will not pay for a traveling companion. If John and Jill are married and both are employees, what is the probability that they will both win?
12. Mutual funds The retirement plan for a company allows employees to invest in 10 different mutual funds. If Sam selected 4 of these funds at random and 6 of the 10 grew by at least 10% over the last year, what is the probability that 3 of Sam’s 4 funds grew by at least 10% last year?
13. Testing In a 10-question matching test with 10 possible answers to match and no answer used more than once, what is the probability of guessing and getting every answer correct?
14E15. Quality control A box of 12 batteries has 3 defective ones. If 2 batteries are drawn from the box together, what is the probability that
(a) both are defective?
(b) neither one is defective?
(c) one is defective?
16. Quality control Suppose that 6 batteries are drawn at random from a box containing 18 good batteries and 2 defective ones. What is the probability that
(a) all 6 are good?
(b) exactly 4 are good?
(c) exactly 2 are good?
17. Quality control A retailer purchases 100 of a new brand of DVD player, of which 2 are defective. The purchase agreement says that if he tests 5 chosen at random and finds 1 or more defective, he receives all the DVD players free of charge. What is the probability that he will not have to pay for the DVD players?
18E19. Banking To see whether a bank has enough minority construction company loans, a social agency selects 30 loans to construction companies at random and finds that 2 of them are loans to minority companies. If the bank’s claim that 10 of every 100 of its loans to construction companies are minority loans is true, what is the probability that 2 loans out of 30 are minority loans? Leave your answer with combination symbols.
20. Diversity A high school principal must select 12 girls at random from the freshman class to serve as hostesses at the junior-senior prom. There are 200 freshman girls, including 20 from minorities, and the principal would like at least one minority girl to have this honor. If he selects the girls at random, what is the probability that
(a) he will select exactly one minority girl?
(b) he will select no minority girls?
(c) he will select at least one minority girl?
21. Diversity Suppose that an employer plans to hire four people from a group of nine equally qualified people of whom three are minority candidates. If the employer does not know which candidates are minority candidates and if she selects her employees at random, what is the probability that
(a) no minority candidates are hired?
(b) all three minority candidates are hired?
(c) one minority candidate is hired?
22. Lottery Four men and three women are semifinalists in a lottery. From this group, three finalists are to be selected by a drawing. What is the probability that all three finalists will be men?
23E24. Sales A car dealer has 12 different cars that he would like to display, but he has room to display only 5.
(a) In how many ways can he pick 5 cars to display?
(b) Suppose 8 of the cars are the same color, with the remaining 4 having distinct colors. If the dealer tells a salesperson to display any 5 cars, what is the probability that all 5 cars will be the same color?
25. Diversity Suppose that two openings on an appellate court bench are to be filled from current municipal court judges. The municipal court judges consist of 23 men and 4 women. Find the probability that
(a) both appointees are men.
(b) one man and one woman are appointed.
(c) at least one woman is appointed.
26E27. Management Suppose that an indecisive company owner has ranked the three top officers of his company at random but claims that they earned their jobs because of ability.
(a) What is the probability that the most able person is ranked at the top?
(b) What is the probability that the top three officers are ranked according to their ability?
28. Quality control Suppose that 10 computer chips are drawn from a box containing 12 good chips and 4 defective chips. What is the probability that
(a) exactly 4 of the chips are defective?
(b) all 10 of the chips are good?
(cl exactly 8 of the chips are good?
29E30E31E32. Evaluation Employees of a firm receive annual reviews. In a certain department, 4 employees received excellent ratings, 15 received good ratings, and 1 received a marginal rating. If 3 employees in this department are randomly selected to complete a form for an internal study of the firm, find the probability that
all 3 selected were rated excellent.
one from each category was selected.
33E34E35. Poker A flush (5 cards from the same suit) is an excellent hand in poker. If 13 cards are in each of four suits in a deck used in the game and 5 cards are in a hand, what is the probability that a flush occurs?
36E37E1CPCan the vectors in Problems 1-4 be probability vectors? If not, why?
2ECan the vectors in Problems 1-4 be probability vectors? If not, why?
3.
Can the vectors in Problems 1-4 be probability vectors? If not, why?
4.
5E6E7E8EIn Problems 9-12, use the given transition matrix and the initial probability vector to find the second probability vector (two steps after the initial probability vector).
10E11E12E13E14E15. Use the matrix and initial probability vector in Problem 11 and find the resulting eighth probability vector.
16EIn Problems 17-20, find the steady-state vector associated with the given transition matrix from Problems 9-12.
17. A
18E19E20EPolitics Use the following information for Problems 21 and 22. In a certain city, the Democratic, Republican, and Consumer parties have members of their parties on the city council. The probability of a member of this party winning any election depends on the proportional membership of his or her party at the time of the election. The probabilities for all these parties winning are given by the following transition matrix P.
21. Using the given transition matrix and assuming that the initial probability vector is , find the probability vectors for the next four steps of the Markov chain. (This initial probability vector indicates that all members are Democrats.)
22EChurch attendance Use the following information for Problems 23-26. The probability that daughters of a mother who attends church regularly will also attend church regularly is 0.8, whereas the probability that daughters of a mother who does not attend regularly will attend regularly is 0.3.
23. What is the transition matrix for this information?
Church attendance Use the following information for Problems 23-26. The probability that daughters of a mother who attends church regularly will also attend church regularly is 0.8, whereas the probability that daughters of a mother who does not attend regularly will attend regularly is 0.3.
25. If a woman does not attend church, what is the probability that her granddaughter attends church regularly?
Church attendance Use the following information for Problems 23-26. The probability that daughters of a mother who attends church regularly will also attend church regularly is 0.8, whereas the probability that daughters of a mother who does not attend regularly will attend regularly is 0.3.
25. If a woman does not attend church, what is the probability that her granddaughter attends church regularly?
26ECar selection Use the following information for Problems 27-32. A man owns an Audi, a Ford, and a VW. He drives every day and never drives the same car two days in a row. These are the probabilities that he drives each of the other cars the next day:
Pr(Ford after Audi) = 0.7 Pr(VW after Audi) = 0.3
Pr(Audi after Ford) = 0.6 Pr(VW after Ford) = 0.4
Pr(Audi after VW) = 0.8 Pr(Ford after VW) = 0.2
27. Write the transition matrix for his selection of a car.
28E29E30E31ECar selection Use the following information for Problems 27-32. A man owns an Audi, a Ford, and a VW. He drives every day and never drives the same car two days in a row. These are the probabilities that he drives each of the other cars the next day:
Pr(Ford after Audi) = 0.7 Pr(VW after Audi) = 0.3
Pr(Audi after Ford) = 0.6 Pr(VW after Ford) = 0.4
Pr(Audi after VW) = 0.8 Pr(Ford after VW) = 0.2
32. Would the probabilities for which car he drives 100 days from now depend on whether he drove a Ford or an Audi today?
33. Population demographics Suppose a government study estimated that the probability of successive generations of a rural family remaining in a rural area was 0.7 and the probability of successive generations of an urban family remaining in an urban area was 0.9. Assuming that a Markov chain applies to these facts, find the steady-state vector.
34E35E36E37E38E39E40E1. If the probability of success on each trial of an experiment is 0.4, what is the probability of 5 successes in 7 trials?
2RE3. If a die is rolled 4 times, what is the probability that a number greater than 4 is rolled at least 2 times?
4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14REDetermine whether each function or table in Problems 15-18 represents a discrete probability distribution.
15.
16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39. Sampling Suppose of a population opposes a proposal and a sample of size 5 is drawn from the population. What is the probability that the majority of the sample will favor the proposal?
40RE41. Disease One person in 100,000 develops a certain disease. Calculate
(a) Pr(exactly 1 person in 100,000 has the disease).
(b) Pr(at least 1 person in 100,000 has the disease).
42RE43RE44RE45RE46. Fraud A company selling substandard drugs to developing countries sold 2,000,000 capsules with 60,000 of them empty (Source: 60 Minutes). If a person gets 100 randomly chosen capsules from this company, what is the expected number of empty capsules this person will get?
47RE48RE49RE50RE51. Net worth Suppose the mean net worth of the residents of Sun City, a retirement community, is $611,000 (Source: The Island Packet). If their net worth is normally distributed with a standard deviation of $96,000,
what percent of the residents have net worths between $700,000 and $800,000?
52. Testing The length of time it takes to complete a college placement test is normally distributed with a mean of 39 minutes and a standard deviation of 6 minutes. How much time (to the nearest minute) is needed so that 90% of the test takers have time to finish.
53RE54RE55RE56RE