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All Textbook Solutions for Mathematical Applications for the Management, Life, and Social Sciences

4E 3. Find the present value of an annuity of $6000 paid at the end of each 6-month period for 8 years if the interest rate is compounded semiannually. 4. Find the present value of an annuity that pays $3000 at the end of each 6-month period for 6 years if the interest rate is compounded semiannually. 5. Suppose a state lottery prize of $5 million is to be paid in 20 payments of $250,000 each at the end of each of the next 20 years. If money is worth compounded annually, what is the present value of the prize? 6. How much is needed in an account that earns compounded monthly to withdraw $1000 at the end of each month for 20 years? 7. With a present value of $135,000, what is the size of the withdrawals that can be made at the end of each quarter for the next 10 years if money is worth 6.4% compounded quarterly? 8. If $88,000 is invested in an annuity that earns 5.8% compounded quarterly, what payments will it provide at the end of each quarter for the next years? 9. A personal account earmarked as a retirement supplement contains $242,400. Suppose $200,000 is used to establish an annuity that earns 6% compounded quarterly and pays $4500 at the end of each quarter. How long will it be until the account balance is $0? 10. A professional athlete invested $2.5 million of a bonus in an account that earns 6.8% compounded semiannually. If $120,000 is to be withdrawn at the end of each six months, how long will it be until the account balance is $0? 11. Suppose that a 25-year government bond has a maturity value of $1000 and a coupon rate of 6%, with coupons paid semiannually. Find the market price of the bond if the yield rate is 5% compounded semiannually. Is this bond selling at a discount or at a premium? 12. Suppose that a 10-year corporate bond has a maturity value of $25,000 and a coupon rate of 7%, with coupons paid semiannually. Find the market price of the bond if the yield rate is 8% compounded semiannually. Is this bond selling at a discount or at a premium? 13. The figure shows a graph that compares the present values of two ordinary annuities of $1000 annually, one at 8% compounded annually and one at 10% compounded annually. Determine which graph corresponds to the 8% rate and which corresponds to the 10% rate. Use the graph to estimate the difference between the present values of these annuities for 25 years. Write a sentence that explains this difference. 14. The figure shows a graph that compares the present values of two ordinary annuities of $800 quarterly, one at 6% compounded quarterly and one at 9% compounded quarterly. Determine which graph corresponds to the 6% rate and which corresponds to the 9% rate. Use the graph to estimate the difference between the present values of these annuities for 25 years (100 quarters). Write a sentence that explains this difference. 17E 18E 17. Find the present value of an annuity due that pays $3000 at the beginning of each quarter for the next 7 years. Assume that money is worth 5.8% compounded quarterly. 18. Find the present value of an annuity due that pays $25,000 every 6 months for the next years if money is worth 6.2% compounded semiannually. 19. What amount must be set aside now to generate payments of $50,000 at the beginning of each year for the next 12 years if money is worth 5.92% compounded annually? 20. Suppose an annuity will pay $15,000 at the beginning of each year for the next 7 years. How much money is needed to start this annuity if it earns 7.3% compounded annually? 21. A year-end bonus of $25,000 will generate how much money at the beginning of each month for the next year, if it can be invested at 6.48% compounded monthly? 22. A couple inherits $89,000. How much can this generate at the beginning of each month over the next 5 years, if money is worth 6.3% compounded monthly? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 23. An insurance settlement of $1.5 million must replace Trixie Eden’s income for the next 40 years. What income will this settlement provide at the end of each month if it is invested in an annuity that earns 8.4% compounded monthly? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 24. A local library receives a bequest of $100,000 from a prominent local family. How much will this provide at the beginning of each 3-month period for the next years if money is worth 7.4% compounded quarterly? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 25. A company wants to have $40,000 at the beginning of each 6-month period for the next years. If an annuity is set up for this purpose, how much must be invested now if the annuity earns 6.68% compounded semiannually? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 26. Is it more economical to buy an automobile for $29,000 cash or to pay $8000 down and $3000 at the end of each quarter for 2 years, if money is worth 8% compounded quarterly? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 27. Dr. Jane Kodiak plans to sell her practice to an HMO. The HMO will pay her $1.5 million now or will make a $500,000 partial payment now and additional payments of $140,000 at the end of each year for the next 10 years. If money is worth 6.5% compounded annually, is it better to take $1.5 million now or $500,000 now and $140,000 at the end of each year for the next 10 years? Justify your choice. In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 28. As a result of a court settlement, an accident victim is awarded $1.2 million. The attorney takes one-third of this amount, another third is used for immediate expenses, and the remaining third is used to set up an annuity. What amount will this annuity pay at the beginning of each quarter for the next 5 years if the annuity earns 7.6% compounded quarterly? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 29. Recent sales of some real estate and record profits make it possible for a manufacturer to set aside $800,000 in a fund to be used for modernization and remodeling. How much can be withdrawn from this fund at the beginning of each half-year for the next 3 years if the fund earns 7.7% compounded semiannually? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 30. A $2.4 million state lottery pays $10,000 at the beginning of each month for 20 years. How much money must the state actually have on hand to set up the payments for this prize if money is worth 6.3% compounded monthly? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 31. How long will an account worth $2.2 million provide $10,000 at the end of each month, if money is worth 5.4% compounded monthly? 34E In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 33. A used piece of rental equipment has years of useful life remaining. When rented, the equipment brings in $800 per month (paid at the beginning of the month). If the equipment is sold now and money is worth 4.8% compounded monthly, what must the selling price be to recoup the income the rental company loses by selling the equipment “early”? 36E In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 35. As the contestant with the longest winning streak in the history of Jeopardy, Ken Jennings won more than $2.5 million. Suppose he invested $1.5 million in an ordinary annuity that earned 7.2% compounded monthly. How much would he receive at the end of each month for the next 20 years? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 36. Suppose Becky has her choice of $10,000 at the end of each month for life or a single prize of $1.5 million. She is 35 years old, and her life expectancy is 40 more years. (i) Find the present value of the annuity if money is worth 7.2% compounded monthly. (ii) If she takes the $1.5 million, spends $700,000 of it, and invests the remainder at 7.2% compounded monthly, what amount will she receive at the end of each month for the next 40 years? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 37. Juanita Domingo’s parents want to establish a college trust for her. They want to make 16 quarterly withdrawals of $2000, with the first withdrawal 3 months from now. If money is worth 7.2% compounded quarterly, how much must be deposited now to provide for this trust? In Problems 23-38, (a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem. 38. A retiree inherits $93,000 and invests it at 6.6% compounded monthly in an annuity that provides an amount at the end of each month for the next 12 years. Find the monthly amount. 39. A 10-year Emmaco Corporate bond has a par value of $10,000 with coupons at 7.8% paid semiannually. If this bond is bought to yield 10% compounded semiannually, find its price. Suppose that after this bond has been held for 2 years, the desired yield is 8% compounded semiannually. Find the selling price. 40. Kodicom Inc. has 15-year bonds with a $5000 maturity value and a quoted coupon rate of 12% paid semiannually. The current yield is 10% compounded semiannually. Compute the price of these bonds. Suppose that with 12 years remaining until maturity, the yield rate drops to 8% compounded semiannually. Find the new price of these bonds. Problems 41-44 are complex financial problems that require several skills, perhaps some from previous sections. For each of Problems 41 and 42, answer the following questions. (a) How much is in the account after the last deposit is made? (b) How much was deposited? (c) What is the amount of each withdrawal? (d) What is the total amount withdrawn? 41. Suppose an individual makes an initial investment of $2500 in an account that earns 7.8% compounded monthly and makes additional contributions of $100 at the end of each month for a period of 12 years. After Problems 41-44 are complex financial problems that require several skills, perhaps some from previous sections. For each of Problems 41 and 42, answer the following questions. (a) How much is in the account after the last deposit is made? (b) How much was deposited? (c) What is the amount of each withdrawal? (d) What is the total amount withdrawn? 42. Suppose that Nam Banh deposits his $12,500 bonus in an account that earns 8% compounded quarterly and makes additional deposits of $500 at the end of each quarter for the next years until he retires. To supplement his retirement, Nam wants to make withdrawals at the end of each quarter for the next 12 years (at which time the account balance will be $0). Problems 41-44 are complex financial problems that require several skills, perhaps some from previous sections. For each of Problems 41 and 42, answer the following questions. (a) How much is in the account after the last deposit is made? (b) How much was deposited? (c) What is the amount of each withdrawal? (d) What is the total amount withdrawn? 43. A young couple wants to have a college fund that will pay $30,000 at the end of each half-year for 8 years. (a) If they can invest at compounded semiannually, how much do they need to invest at the end of each 6-month period for the next 18 years to begin making their college withdrawals 6 months after their last investment? (b) Suppose 8 years after beginning the annuity payments, they receive an inheritance of $38,000 that they contribute to the account, and they continue to make their regular payments as found in part (a). How many college withdrawals will they be able to make before the account balance is $0? Problems 41-44 are complex financial problems that require several skills, perhaps some from previous sections. For each of Problems 41 and 42, answer the following questions. (a) How much is in the account after the last deposit is made? (b) How much was deposited? (c) What is the amount of each withdrawal? (d) What is the total amount withdrawn? 44. A recent college graduate begins a savings plan at age 27 by investing $400 at the end of each month in an account that earns compounded monthly. (a) If this plan is followed for 10 years, how much should the monthly contributions be for the next 28 years to be able to withdraw $10,000 at the end of each month from the account for the next 25 years? (b) What is the total amount contributed? (c) What is the total amount withdrawn? 47E 46. Find the present value of an annuity of $2000 per year at the end of each of 8 years after being deferred for 6 years, if money is worth 7% compounded annually. 47. The terms of a single parent’s will indicate that a child will receive an ordinary annuity of $16,000 per year from age 18 to age 24 (so that the child can attend college) and that the balance of the estate goes to a niece. If the parent dies on the child’s 14th birthday, how much money must be removed from the estate to purchase the annuity? (Assume an interest rate of 6% compounded annually.) 48. On his 48th birthday, a man wants to set aside enough money to provide an income of $1500 at the end of each month from his 60th birthday to his 65th birthday. If he earns 6% compounded monthly, how much will this supplemental retirement plan cost him on his 48th birthday? 49. The semiannual tuition payment at a major university is expected to be $30,000 for the 4 years beginning 18 years from now. What lump sum payment should the university accept now in lieu of tuition payments beginning 18 years, 6 months from now? Assume that money is worth 7% compounded semiannually and that tuition is paid at the end of each half-year for 4 years. 50. A community is soliciting pledges for a fitness center project to begin in 3 years. If money is worth 6% compounded quarterly, how much must a family deposit now to contribute $2500 at the end of each quarter for years after the project begins? 51. Danny Metzger’s parents invested $ 1600 when he was born. This money is to be used for Danny’s college education and is to be withdrawn in four equal annual payments beginning when Danny is age 19. Find the amount that will be available each year, if money is worth 6% compounded annually. 52. Carol Goldsmith received a trust fund inheritance of $10,000 on her 30th birthday. She plans to use the money to supplement her income with 20 quarterly payments beginning on her 60th birthday. If money is worth 7.6% compounded quarterly, how much will each quarterly payment be? 55E 54. A couple received a $134,000 inheritance the year they turned 48 and invested it in a fund that earns 7.7% compounded semiannually. If this amount is deferred for 14 years (until they retire), how much will it provide at the end of each half-year for the next 20 years after they retire? CHECKPOINT 1. A debt of $25,000 is to be amortized with equal quarterly payments over 6 years, and money is worth 7% compounded quarterly. (a) Find the total number of payments. (b) Find the interest rate per period. (c) Write the formula used to find the size of each payment. (d) Find the amount of each payment. CHECKPOINT 2. A new college graduate determines that she can afford a car payment of $400 per month. If the auto manufacturer is offering a special 2.1% financing rate compounded monthly for 5 years, how much can she borrow and still have a $400 monthly payment? 3CP 1E 2E 3. A debt of $8000 is to be amortized with 8 equal semiannual payments. If the interest rate is compounded semiannually, what is the size of each payment? 4. A loan of $10,000 is to be amortized with 10 equal quarterly payments. If the interest rate is compounded quarterly, what is the periodic payment? 5. A recent graduate’s student loans total $18,000. If these loans are at compounded quarterly for 10 years, what are the quarterly payments? 6. For equipment upgrades, a business borrowed $400,000 at 8% compounded semiannually for 5 years. What are the semiannual payments? 7. A homeowner planning a kitchen remodeling can afford a $600 monthly payment. How much can the homeowner borrow for 5 years at 6% compounded monthly and still stay within the budget? 8. AdriAnne and Anna’s Auto Repair wants to add a new service bay. How much can they borrow at 5% compounded quarterly for years, if the desired quarterly payment is $10,000? 9E In Problems 9-12, develop an amortization schedule for the loan described. 12. $50,000 for 5 years at compounded semiannually 11E 12E 13. A $10,000 loan is to be amortized for 10 years with quarterly payments of $334.27. If the interest rate is 6% compounded quarterly, what is the unpaid balance immediately after the sixth payment? 14. A debt of $8000 is to be amortized with 8 equal semiannual payments of $1288.29. If the interest rate is 12% compounded semiannually, find the unpaid balance immediately after the fifth payment. 15. When Maria Acosta bought a car years ago, she borrowed $28,000 for 48 months at 8.1% compounded monthly. Her monthly payments are $684.88, but she’d like to pay off the loan early. How much will she owe just after her payment at the -year mark? 16E Problems 17-20 describe a debt to be amortized. In each problem, find: (a) the size of each payment, (b) the total amount paid for each purchase, (c) the total interest paid over the life of the loan. 17. A man buys a house for $350,000. He makes a $150,000 down payment and amortizes the rest of the purchase price with semiannual payments over the next 10 years. The interest rate on the debt is 12% compounded semiannually. Problems 17-20 describe a debt to be amortized. In each problem, find: (a) the size of each payment, (b) the total amount paid for each purchase, (c) the total interest paid over the life of the loan. 18. Sean Lee purchases $20,000 worth of supplies for his restaurant by making a $3000 down payment and amortizing the remaining cost with quarterly payments over the next 5 years. The interest rate on the debt is 16% compounded quarterly. Problems 17-20 describe a debt to be amortized. In each problem, find: (a) the size of each payment, (b) the total amount paid for each purchase, (c) the total interest paid over the life of the loan. 19. A woman buys an apartment house for $1,250,000 by making a down payment of $250,000 and amortizing the rest of the purchase price with monthly payments over the next 10 years. The interest rate on the debt is 7.2% compounded monthly. Problems 17-20 describe a debt to be amortized. In each problem, find: (a) the size of each payment, (b) the total amount paid for each purchase, (c) the total interest paid over the life of the loan. 20. John Fare purchased $10,000 worth of equipment by making a $2000 down payment and promising to pay the remainder of the cost in semiannual payments over the next 4 years. The interest rate on the debt is 10% compounded semiannually. 21. A man buys a car for $36,000. If the interest rate on the loan is 12% compounded monthly and he wants to make monthly payments of $900 for 36 months, how much must he put down? 22. A woman buys a car for $40,000. If the interest rate on the loan is 12% compounded monthly and she wants to make monthly payments of $700 for 3 years, how much must she have for a down payment? 23. A couple purchasing a home budget $1800 per month for their loan payment. If they have $20,000 available for a down payment and are considering a 25-year loan, how much can they spend on the home at each of the following rates? (a) 6.9% compounded monthly (b) 7.5% compounded monthly 24. A developer wants to buy a certain parcel of land. The developer believes she can afford payments of $44,000 each half-year for the next 7 years. How much can she borrow and hold to this budget at each of the following interest rates? (a) 8.9% compounded semiannually (b) 7.3% compounded semiannually 25. A couple who borrow $90,000 for 30 years at 7.2% compounded monthly must make monthly payments of $610.91. Find their unpaid balance after 1 year. During that first year, how much interest do they pay? 26. A company that purchases a piece of equipment by borrowing $250,000 for 10 years at 6% compounded monthly has monthly payments of $2775.51. Find the unpaid balance on this loan after 1 year. During that first year, how much interest does the company pay? 27. When Otto and Millie bought their home, they borrowed $200,000 for 30 years at 6% compounded monthly. After making 120 payments of $1199.10, they plan to refinance at 4.5% compounded monthly for 15 years, with refinancing costs of $750 added to the amount of the new loan. Find the amount of the new loan (amount refinanced). Find their new monthly payment. Find the amount saved by refinancing. 28E A recent college graduate buys a new car by borrowing $18,000 at 8.4%, compounded monthly, for 5 years. She decides to pay an extra $15 per payment. (a) What is the monthly payment required by the loan, and how much does she decide to pay each month? (b) How many payments (that include the extra $15) will she make? (c) How much will she save by paying the extra $15?A young couple buying their first home borrow $85,000 for 30 years at 7.2%, compounded monthly, and make payments of $576.97. After 3 years, they are able to make a one-time payment of $2000 along with their 36th payment. (a) Find the unpaid balance immediately after they pay the extra $2000 and their 36th payment. (b) How many regular payments of $576.97 will amortize the unpaid balance from part (a)? (c) How much will the couple save over the life of the loan by paying the extra $2000?31. Jadele, Inc., borrowed $12.8 million at 7.2% compounded quarterly for 30 years for construction of a new manufacturing facility. After making 42 quarterly payments of $261,094.80, it plans to refinance the existing loan for an amount that includes an additional $1.1 million for expansion. Jadele can refinance at 6.6% compounded quarterly for 25 years, with refinancing costs of $10,000 added to the refinanced amount. Find the total amount refinanced. Find the new quarterly payment. If Jadele decides to continue to pay $261,094.80 per quarter on the refinanced amount, how many payments will it take to pay off the new loan? 32. When Gustavo and Serrana bought their home, they had a 5.7% loan with monthly payments of $870.60 for 30 years. After making 78 monthly payments, they plan to refinance for an amount that includes an additional $35,000 to remodel their kitchen. They can refinance at 4.8% compounded monthly for 25 years, with refinancing costs of $625 included with the amount refinanced. Find the amount refinanced. Find their new monthly payment. How long will it take to pay off this new loan if they pay $1200 each month? 33E 34E 35E 36. Some banks now have biweekly mortgages (that is, with payments every other week). Compare a 20-year, $100,000 loan at 8.1% by finding the payment size and the total interest paid over the life of the loan under each of the following conditions. (a) Payments are monthly, and the rate is 8.1 % compounded monthly. (b) Payments are biweekly, and the rate is 8.1 % compounded biweekly. 37E 38. Time-share sales provide an opportunity for vacationers to own a resort condo for 1 week (or more) each year forever. The owners may use their week at their own condo or trade the week and vacation elsewhere. Time-share vacation sales usually require payment in full or financing through the time-share company, and interest rates are usually in the 13%-18% range. Suppose the cost to buy a 1-week time share in a 3-bed- room condo is $21,833. Also suppose a 10% down payment is required, with the balance financed for 15 years at 16.5% compounded monthly. (a) Find the monthly payment. (b) Determine the total cost over the life of the loan. (c) Suppose maintenance fees for this condo are $400 per year. Find the annual cost of the condo over the life of the loan. Assume that the annual maintenance fees remain constant. (d) Use part (c) and the 10% down payment to determine the average annual cost for having this vacation condo for 1 week over the life of the loan. COMBINED APPLICATIONS Problems 39 and 40 are complex financial problems that require several skills, perhaps some from previous sections. During four years of college, Nolan MacGregors student loans are $4000, $3500, $4400, and $5000 for freshman year through senior year, respectively. Each loan amount gathers interest of 1%, compounded quarterly, while Nolan is in school and 3%, compounded quarterly, during a 6-month grace period after graduation. (a) What is the loan balance after the grace period? Assume the freshman year loan earns 1% interest for 3/4 year during the first year, then for 3 full years until graduation. Make similar assumptions for the loans for the other years. (b) After the grace period, the loan is amortized over the next 10 years at 3%, compounded quarterly. Find the quarterly payment. (c) If Nolan decides to pay an additional $90 per payment, how many payments will amortize the debt? (d) How much will Nolan save by paying the extra $90 with each payment?COMBINED APPLICATIONS Problems 39 and 40 are complex financial problems that require several skills, perhaps some from previous sections. Clark and Lana take a 30-year home mortgage of $121,000 at 7.8%, compounded monthly. They make their regular monthly payments for 5 years, then decide to pay $1000 per month. (a) Find their regular monthly payment. Clark and Lana take a 30-year home mortgage of $121,000 at 7.8%, compounded monthly. They make their regular monthly payments for 5 years, then decide to pay $1000 per month. (a) Find their regular monthly payment.1. If one ball is drawn from a bag containing 9 balls numbered 1 through 9, what is the probability that the ball’s number is (a) odd? (b) divisible by 3? (c) odd and divisible by 3? 2. Suppose a ball is drawn from a bag containing 9 red balls numbered 1 through 9 and 3 white balls numbered 10 through 12. Two possible sample spaces for this event are and . (a) Which of these sample spaces is equiprobable? What is the probability that (b) the ball is red? (c) the ball is odd-numbered? (d) the ball is white and even-numbered? (e) the ball is white or odd-numbered? 3. If the probability that an event E occurs is 3/7, what are the odds that (a) E will occur? (b) E will not occur? 4. Suppose that a fair coin is tossed two times. Construct an equiprobable sample space for the experiment and determine each of the following probabilities. (a) Pr(0 heads) (b) Pr(l head) (c) Pr(2 heads) 5. Suppose that a fair coin is tossed three times. Construct an equiprobable sample space for the experiment and determine each of the following probabilities. (a) Pr(2 heads) (b) Pr(3 heads) (c) Pr( 1 head) 6. A card is drawn at random from an ordinary deck of 52 playing cards. What is the probability that it is a queen or a jack? 7. A deck of 52 cards is shuffled. A card is drawn, it is replaced, the pack is shuffled again, and a second card is drawn. What is the probability that each card drawn is an ace, a king, a queen, or a jack? 8RE 9RE 10RE 11. A bag contains 4 red balls numbered 1, 2, 3, 4 and 5 white balls numbered 5, 6, 7, 8, 9. A ball is drawn. What is the probability that the ball (a) is red and even-numbered? (b) is red or even-numbered? (c) is white or odd-numbered? 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19. Bag A contains 3 red, 6 black, and 5 white balls, and bag B contains 4 red, 5 black, and 7 white balls. A bag is selected at random, and a ball is drawn. If the ball is white, what is the probability that bag B was selected? 20RE 21RE 22RE 23. How many combinations of 8 things taken 5 at a time are possible? 24RE 25RE 26. Find the steady-state vector associated with the transition matrix. 27. Senior citizens In a certain city, 30,000 citizens out of 80,000 are over 50 years of age. What is the probability that a citizen selected at random will be 50 years old or younger? 28RE 29. United Nations Of 100 job applicants to the United Nations, 30 speak French, 40 speak German, and 12 speak both French and German. If an applicant is chosen at random, what is the probability that the applicant speaks French or German? 30RE 31RE Productivity Suppose that in a study of leadership style versus industrial productivity, the following data were obtained. Use these empirical data to answer Problems 30-32. Leadership Style Productivity Democratic Authoritarian Laissez- faire Total Low 40 15 40 95 Medium 25 75 10 110 High 25 30 20 75 Total 90 120 70 280 32. Find the probability that an individual chosen at random has an authoritarian style given that he or she has medium productivity. 33. Quality control A product must pass an initial inspection, where the probability that it will be rejected is 0.2. If it passes this inspection, it also must pass a second inspection where the probability that it will be rejected is 0.1. What is the probability that it will pass both inspections? 34RE 36. Purchasing A regional survey found that of all families who indicated an intention to buy a new car bought a new car within 3 months, that of families who did not indicate an intention to buy a new car bought one within 3 months, and that indicated an intention to buy a new car. If a family chosen at random bought a car, find the probability that the family had not previously indicated an intention to buy a car. 38. Management A personnel director ranks 4 applicants for a job. How many rankings are possible? 37RE 38RE 39RE 40RE 43. License plates The RBC Heritage Classic Foundation license plate is available to anyone with a vehicle registered in the state of South Carolina. It costs $75 every two years with $66-$71 going to the foundation for its Scholar and Birdies for Charity Programs. The plate has the Heritage Classic symbol and four spaces for nonzero digits and/or letters. How many possible plates can be produced using from 1 to 4 digits or letters if they can be repeated? 42RE 43RE 44RE 45RE 48. Quality control A supplier has 200 compact discs, of which are known to be defective. If a music store purchases 10 of these discs, what is the probability that (a) none of the discs is defective? (b) 2 of the discs are defective? 49. Stocks Mr. Way must sell stocks from 3 of the 5 companies whose stocks he owns so that he can send his children to college. If he chooses the companies at random, what is the probability that the 3 companies will be the 3 with the best future earnings? 50. Quality control A sample of 6 fuses is drawn from a lot containing 10 good fuses and 2 defective fuses. Find the probability that the number of defective fuses is (a) exactly 1. (b) at least 1. Income levels Use the following information for Problems 51 and 52. Suppose people in a certain community are classified as being low-income (L), middle- income (M), or high-income (H). Suppose further that the probabilities for children being in a given state depend on which state their parents were in according to the following Markov chain transition matrix. 51. If the initial distribution probabilities of families in a certain population are , find the distribution for the next three generations. 50RE 1T 2T A bag contains three white balls numbered 1, 2, 3 and four black balls numbered 4, 5, 6, 7. If one ball is drawn at random, find the probability of each event described in Problems 1-3. 3. (a) The ball is red. (b) The ball is numbered less than 8. A bag contains three white balls numbered 1, 2, 3 and four black balls numbered 4, 5, 6, 7. Two balls are drawn without replacement. Find the probability of each event described in Problems 4-9. 4. The sum of the numbers is 7. A bag contains three white balls numbered 1, 2, 3 and four black balls numbered 4, 5, 6, 7. Two balls are drawn without replacement. Find the probability of each event described in Problems 4-9. 5. Both balls are white. 6T 7T 8T 9T 10. A cat hits the letters on a computer keyboard three times. What is the probability that the word RAT appears? 11T 12T 13T 14T 15T 16. Computer chips come from two suppliers, with coming from supplier 1 and coming from supplier 2. Six percent of the chips from supplier 1 are defective, and of the chips from supplier 2 are defective. If a chip is chosen at random, what is the probability that it is defective? 17. A placement test is given by a university to predict student success in a calculus course. On average, 70% of students who take the test pass it, and 87% of those who pass the test also pass the course, whereas 8% of those who fail the test pass the course. (a) What is the probability that a student taking the placement test and the calculus course will pass the course? (b) If a student passed the course, what is the probability that he or she passed the test? 18. Lactose intolerance affects about of non-Hispanic White Americans and 50% of White Hispanic Americans. Nine percent of Americans are Hispanic Whites, and are non-Hispanic Whites (Source: Jean Carper, “Hat Smart,” USA Weekend). If a White American is chosen at random and is lactose-intolerant, what is the probability that be or she is Hispanic? 19T 20T 21T 22. A publishing company has determined that a new edition of an existing mathematics textbook will be readopted by 80% of its current users and will be adopted by 7% of the users of other texts if the text is not changed radically. To determine whether it should change the book radically to attract more sales, the company uses Markov chains. Assume that the text in question currently has 25% of its possible market. (a) Create the transition matrix for this chain. (b) Find the probability vector for the text three editions later and, from that, determine the percent of the market for that future edition. (c) Find the steady-state vector for this text to determine what percent of its market this text will have if this policy is continued. CHECKPOINT 1. If a coin is drawn from a box containing 4 gold, 10 silver, and 16 copper coins, all the same size, what is the probability of (a) getting a gold coin? (b) getting a copper coin? (c) not getting a copper coin? CHECKPOINT 2. A ball will be drawn from a bag containing balls numbered 1, 2, 3, 4, and 5. To find the probability that a ball with an even number is drawn, we can use the sample space . (a) Which of these sample spaces is an equiprobable sample space? (b) What is the probability of drawing a ball that is even-numbered? CHECKPOINT 3. A coin is tossed, and a die is rolled. (a) Write an equiprobable sample space listing the possible outcomes for this experiment. (b) Find the probability of getting a head on the coin and an even number on the die. (c) Find the probability of getting a tail on the coin and a number divisible by 3 on the die. 1. One ball is drawn at random from a bag containing 4 red balls and 6 white balls. What is the probability that the ball is (a) red? (b) green? (c) red or white? 2. One ball is drawn at random from a bag containing 4 red balls and 6 white balls. What is the probability that the ball is (a) white? (b) white or red? (c) red and white? 3. If you draw one card at random from a deck of 12 cards numbered 1 through 12, inclusive, what is the probability that the number you draw is divisible by 4? 4E 5E 6. A die is rolled. What is the probability that (a) a 4 will result? (b) a 7 will result? (c) an odd number will result? 7. An urn contains three red balls numbered 1, 2, 3, four white balls numbered 4, 5, 6, 7, and three black balls numbered 8, 9, 10. A ball is drawn from the urn. What is the probability that (a) it is red? (b) it is odd-numbered? (c) it is red and odd-numbered? (d) it is red or odd-numbered? (e) it is not black? 8. An urn contains three red balls numbered 1, 2, 3, four white balls numbered 4, 5, 6, 7, and three black balls numbered 8, 9, 10. A ball is drawn from the urn. What is the probability that the ball is (a) white? (b) white and odd? (c) white or even? (d) black or white? (e) black and white? 9. From a deck of 52 ordinary playing cards, one card is drawn. Find the probability that it is (a) a queen. (b) a red card. (c) a spade. 10. From a deck of 52 ordinary playing cards, one card is drawn. Find the probability that it is (a) a red king. (b) a king or a black card. 11. Suppose a fair coin is tossed two times. Construct an equiprobable sample space for the experiment and determine each of the following probabilities. (a) Pr(0 heads) (b) Pr( 1 head) (c) Pr(2 heads) 12E 13. Use Table 7.1 to determine the following probabilities if a distinguishable pair of dice is rolled. (a) Pr(sum is 4) (b) Pr(sum is 10) (c) Pr(sum is 12) To find the probability of obtaining a given sum when a pair of dice is rolled, we need to determine the outcomes. However, the sums {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} are not equally likely; there is only one way to obtain a sum of 2, but there are several ways to obtain a sum of 6. If we distinguish between the two dice we are rolling and we record all the possible outcomes for each die, we see that there are 36 possibilities, each of which is equally likely. Table 7.1 shows these possibilities, with the event “sum = 9" encircled. TABLE 7.1 This list of possible outcomes for finding the sum of two dice is an equiprobable sample space for the experiment. Because the 36 elements in the sample space are equally likely, we can find the probability that a given sum results by determining the number of ways this sum can occur and dividing that number by 36. 14. (a) When a pair of distinguishable dice is rolled, what sum is most likely to occur? (b) When a pair of distinguishable dice is rolled, what is Pr(), where S represents the sum rolled? 15E 16E 17. Suppose a die is tossed 1200 times and a 6 comes up 431 times. (a) Find the empirical probability for a 6 to occur. (b) On the basis of a comparison of the empirical probability and the theoretical probability, do you think the die is fair or biased? 18. Suppose that a coin is tossed 3000 times and 1800 heads result. What is the empirical probability that a head will occur with this coin? Is there evidence that the coin is a fair coin? 19E 20E 21. If the odds that a particular horse will win a race are , what is the probability (a) that the horse will win the race? (b) that the horse will lose the race? 22. If the odds that an event E will not occur are , what is the probability (a) that event E will occur? (b) that event E will not occur? 23E 24E 25. Car maintenance A car rental firm has 425 cars. Sixty- three of these cars have defective turn signals, and 32 have defective tires. What is the probability that one of these cars selected at random (a) has defective turn signals? (b) has defective tires? 26E 27. Voting The table gives the average number of voters in Springfield in each of three political parties during the last 12 years, along with the average number that voted in presidential elections during this period. (a) For each political party, use these data to find the probability that a person selected at random from the registered voters in the party will vote in the next election. (b) For which party is the probability highest? Political Party Republican Democratic Independent Registered voters 4500 6100 2200 Voted 2835 2501 1122 28E 29. Sales promotion In a sales promotion, a clothing store gives its customers a chance to draw a ticket from a box that contains a discount on their next purchase. The box contains 3000 tickets giving a 10% discount, 500 giving a 30% discount, 100 giving a 50% discount, and 1 giving a 100% discount. What is the probability that a given customer will randomly draw a ticket giving (a) a 100% discount? (b) a 50% discount? (c) a discount of less than 50%? (d) Is a given customer more likely to get a 30% discount or a discount higher than 30%? 30. Management A dry cleaning firm has 12 employees: 7 women and 5 men. Three of the women and five of the men are 40 years or older. The remainder are over 20 and under 40. If a person is chosen at random from this firm, what is the probability that the person is (a) a woman? (b) under 40 years of age? 31E 32E 33. Blood types Human blood is classified by blood type, which indicates the presence or absence of the antigens A, B, and Rh, as follows. A present Type A B present Type B Both A and B present Type AB Neither A nor B present Type O Each of these types is combined with a + or – sign to indicate whether the Rh antigen is present. Write a sample space containing all possible blood types. Do you think this is an equiprobable sample space? Explain. 34E 35E 36E 37E 38E 39. Fraud A company selling substandard drugs to developing countries sold 2,000,000 capsules with 60,000 of them empty (Source; 60 Minutes). What is the probability that a person who takes a randomly chosen capsule from this company will get an empty capsule? 40E 41E 42E 43E 44. Quality control A supply of 500 plasma television displays has 6 defective displays. What is the probability that a display picked at random from the supply is not defective? 45E 46E 47E 48. Quality control A computer store offers used computers free to local middle schools. Of the 54 machines available, four have defective memories, six others have defective keys, and the remainder have no defects. If a teacher picks one at random, what is the probability that she will select a defective computer? 49. Genetics A newly married couple plan to have three children. Assuming that the probability of a girl being born equals that of a boy being born, what is the probability that exactly two of the three children born will be girls? (Hint: Construct the sample space.) 50. Management A frustrated store manager is asked to make four different yes-no decisions that have no relation to each other. Because he is impatient to leave work, he flips a coin for each decision. If the correct decision in each case was yes, what is the probability that all of his decisions were correct? none of his decisions was correct? half of his decisions were correct? 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64. Ice cream The odds of an American home having at least one container of ice cream in the freezer are 9 to 10. What is the probability that an American home chosen at random will have ice cream in its freezer? 65E 66. Rain If the probability of rain today is , what are the odds against rain today? 67E 68E CHECKPOINT 1. If a ball is drawn from a bag containing 4 red balls numbered 1, 2, 3,4 and 3 white bulls numbered 5, 6, 7, what b the probability that the ball b (a) red and even? (b) white and even? (c) not red 2CP 1E 2E 5. An ordinary die is tossed. What is the probability of getting a number divisible by 3 or an odd number? 6. A card is drawn at random from a deck of 52 playing cards. Find the probability that it is a club or a king. 7. If the probability that event E will occur is 3/5, what is the probability that E will not occur? 6E In Problems 7-8, a bag contains 5 red balls numbered 1, 2, 3,4, 5 and 9 white balls numbered 6, 7,8,9, 10, 11, 12, 13, 14. If a ball is drawn, what is the probability that (a) the ball is red and even-numbered? (b) the ball is red or even-numbered?8E 9E 13. Suppose a die is biased (unfair) so that each odd-numbered face has probability 1/4 of resulting and each even face has probability 1/12 of resulting. Find the probability of getting a number greater than 3. 11E 12E 15. A bag contains 4 white, 7 black, and 6 green balls. What is the probability that a ball drawn at random from the bag is white or green? 16. In a game where only one player can win, the probability that Jack will win is 1/5 and the probability that Bill will win is 1/4. Find the probability that one of them will win. 17. A cube has 2 faces painted red, 2 painted white, and 2 painted blue. What is the probability of getting a red face or a white face in one roll? 16E 19. A ball is drawn from a bag containing 13 red balls numbered 1-13 and 5 white balls numbered 14-18. What is the probability that the ball is not even-numbered? red and even-numbered? red or even-numbered? neither red nor even-numbered? 20. A ball is drawn from a bag containing 13 red balls numbered 1-13 and 5 white balls numbered 14-18. What is the probability that the ball is not red? white and odd-numbered? white or odd-numbered? neither white nor odd-numbered? 21. Drug use Forty-six percent of marijuana use among youth occurs in the inner cities (Source: Partnership for a Drug-Free America). If an instance of such marijuana use is chosen at random, what is the probability that the use does not occur in an inner city? 22. Breast cancer According to the American Cancer Society, 199 of 200 mammograms turn out to be normal. What is the probability that the mammogram of a woman chosen at random will show an abnormality? 23. Car maintenance A car rental firm has 425 cars. Sixty-three of these cars have defective turn signals, and 32 have defective tires. What is the probability that one of these cars selected at random does not have defective turn signals? has no defects if no car has 2 defects? 25. Linguistics Of 100 students, 24 can speak French, 18 can speak German, and 8 can speak both French and German. If a student is picked at random, what is the probability that he or she can speak French or German? 25. Linguistics Of 100 students, 24 can speak French, 18 can speak German, and 8 can speak both French and German. If a student is picked at random, what is the probability that he or she can speak French or German? 26. Management A company employs 65 people. Eight of the 30 men and 21 of the 35 women work in the business office. What is the probability that an employee picked at random is a woman or works in the business office? 25E Photography Rob Lee knows his camera will take a good picture unless the flash is defective or the batteries are dead. The probability of having a defective flash is 0.05, the probability of the batteries being dead is 0.3, and the probability that both these problems occur is 0.01. What is the probability that the picture will be good?29. Salaries The table gives the percent of employees of the Ace Company in each of three salary brackets categorized by the sex of the employees. An employee is selected at random. (a) What is the probability that the person selected is female and makes less than $60,000? (b) What is the probability that the person selected makes at least $100,000? (c) What is the probability that the person is male or makes less than $60,000? Earns Earns at Least Earns Less Than $60,000 and Less at Least $60,000 Than $100,000 $100,000 Male 25% 18% 5% Female 35% 14% 3% 28E 31. Workers The table gives the projected number of workers in various categories in Springfield in 2020. If one of the represented workers is chosen at random, use the table to find the probability that the person (a) is Latino or female. (b) is male or Black. (c) is Asian or White. White Black Asian Latino Male 49,804 6084 2780 10,044 Female 36,026 6664 2078 5671 Total 85,830 12,748 4858 15,715 30E 31E 32E 33E 34E 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. 38E 39E 40E 41E 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? 43E 44E 45E 46E CHECKPOINT 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? 2CP 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 2E 3. 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? 6E 7. 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?

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