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decided? (Assume that there is no tie.) 7. Multiple-Choice Test A multiple-choice test has five questions with four choices for each question. In how many different ways can the test be completed? 8. Phone Numbers Telephone numbers consist of seven digits; the first digit cannot be 0 or 1. How many telephone numbers are possible? 9. Running a Race In how many different ways can a race with five runners be completed? (Assume that there is no tie.) 10. Seating Order In how many ways can five people be seated in a row of five seats? 11. Restaurant Meals A restaurant offers the items listed in the table. How many different meals consisting of a main course, a drink, and a dessert can be selected at this restaurant? Main courses Drinks Desserts Chicken Beef Lasagna Quiche Ice cream Layer cake Blueberry pie Iced tea Apple juice Cola Ginger ale Coffee 12. Lining Up Books In how many ways can five different mathematics books be placed next to each other on a shelf? 13. Multiple Routes Towns A, B, C, and D are located in such a way that there are four roads from A to B, five roads from B to C, and six roads from C to D. How many routes are there from town A to town D via towns B and C? 14. Birth Order In a family of four children, how many different boy-girl birth-order combinations are possible? (The birth orders BBBG and BBGB are different.) 15. Flipping a Coin A coin is flipped five times, and the result- ing sequence of heads and tails is recorded. How many such sequences are possible? 16. Rolling a Pair of Dice A red die and a white die are rolled, and the numbers that show are recorded. How many different outcomes are possible? (The singular form of the word dice is die.) 17. Rolling Three Dice A red die, a blue die, and a white die are rolled, and the numbers that show are recorded. How many different outcomes are possible? 18. Picking Cards Two cards are chosen in order from a deck. In how many ways can this be done if (a) the first card must be a spade and the second must be a heart? (b) both cards must be spades? 19. Choosing Out |
fits A girl has 5 skirts, 8 blouses, and 12 pairs of shoes. How many different skirt-blouse-shoe outfits can she wear? (Assume that each item matches all the others, so she is willing to wear any combination.) 20. ID Numbers A company’s employee ID number system consists of one letter followed by three digits. How many different ID numbers are possible with this system? 21. ID Numbers A company has 2844 employees. Each em- ployee is to be given an ID number that consists of one letter followed by two digits. Is it possible to give each employee a different ID number using this scheme? Explain. 22. Pitchers and Catchers An all-star baseball team has a roster of seven pitchers and three catchers. How many pitcher-catcher pairs can the manager select from this roster? ✎ 23. License Plates Standard automobile license plates in California display a nonzero digit, followed by three letters, followed by three digits. How many different standard plates are possible in this system? 24. Combination Lock A combination lock has 60 different positions. To open the lock, the dial is turned to a certain number in the clockwise direction, then to a number in the counterclockwise direction, and finally to a third number in the clockwise direction. If successive numbers in the combination SE CTI O N 10.1 | Counting Principles 661 cannot be the same, how many different combinations are possible? 25. True-False Test A true-false test contains ten questions. In how many different ways can this test be completed? 26. Ordering a Car An automobile dealer offers five models. Each model comes in a choice of four colors, three types of stereo equipment, with or without air conditioning, and with or without a sunroof. In how many different ways can a customer order an auto from this dealer? 27. Classifications The registrar at a certain university classifies students according to a major, minor, year (1, 2, 3, 4), and sex (M, F). Each student must choose one major and either one or no minor from the 32 fields taught at this university. How many different student classifications are possible? 28. Monograms How many monograms consisting of three ini- tials are possible? 29. License Plates |
A state has registered 8 million automobiles. To simplify the license plate system, a state employee suggests that each plate display only two letters followed by three digits. Will this system create enough different license plates for all the vehicles that are registered? 30. License Plates A state license plate design has six places. Each plate begins with a fixed number of letters, and the remaining places are filled with digits. (For example, one letter followed by five digits, two letters followed by four digits, and so on.) The state has 17 million registered vehicles. (a) The state decides to change to a system consisting of one letter followed by five digits. Will this design allow for enough different plates to accommodate all the vehicles that are registered? (b) Find a system that will be sufficient if the smallest possible number of letters is to be used. 31. Class Executive In how many ways can a president, vice president, and secretary be chosen from a class of 30 students? 32. Class Executive In how many ways can a president, vice president, and secretary be chosen from a class of 20 females and 30 males if the president must be a female and the vice president must be a male? 33. Committee Officers A senate subcommittee consists of ten Democrats and seven Republicans. In how many ways can a chairman, vice chairman, and secretary be chosen if the chairman must be a Democrat and the vice chairman must be a Republican? 34. Social Security Numbers Social Security numbers consist of nine digits, with the first digit between 0 and 6, inclusive. How many Social Security numbers are possible? 662 CHAPTER 10 | Counting and Probability 35. Five-Letter Words Five-letter “words” are formed using the letters A, B, C, D, E, F, G. How many such words are possible for each of the following conditions? (a) No condition is imposed. (b) No letter can be repeated in a word. (c) Each word must begin with the letter A. (d) The letter C must be in the middle. (e) The middle letter must be a vowel. 36. Palindromes How many five-letter palindromes are pos- sible? (A palindrome is a string of letters that reads the same backward and forward, such as the string XCZCX.) 37. Names of |
Variables A certain computer programming language allows names of variables to consist of two characters, the first being any letter and the second being any letter or digit. How many names of variables are possible? 38. Code Words How many different three-character code words consisting of letters or digits are possible for the following code designs? (a) The first entry must be a letter. (b) The first entry cannot be zero. 39. Seating Arrangements In how many ways can four men and four women be seated in a row of eight seats for the following situations? (a) The women are to be seated together, and the men are to be seated together. (b) They are to be seated alternately by gender. 40. Arranging Books In how many ways can five different mathematics books be placed on a shelf if the two algebra books are to be placed next to each other? 41. Arranging Books Eight mathematics books and three chem- istry books are to be placed on a shelf. In how many ways can this be done if the mathematics books are next to each other and the chemistry books are next to each other? 42. Three-Digit Numbers Three-digit numbers are formed using the digits 2, 4, 5, and 7, with repetition of digits allowed. How many such numbers can be formed if (a) the numbers are less than 700? (b) the numbers are even? (c) the numbers are divisible by 5? 43. Three-Digit Numbers How many three-digit odd numbers can be formed using the digits 1, 2, 4, and 6 if repetition of digits is not allowed? ▼ DISCOVE RY • DISCUSSION • WRITI NG 44. Pairs of Initials Explain why in any group of 677 people, at least two people must have the same pair of initials. 45. Area Codes Until recently, telephone area codes in the United States, Canada, and the Caribbean islands were chosen according to the following rules: (i) The first digit cannot be 0 or a 1, and (ii) the second digit must be a 0 or a 1. But in 1995 the second rule was abandoned when the area code 360 was introduced in parts of western Washington State. Since then, many other new area codes that violate Rule (ii) have come into use, although Rule (i) still remains in effect. (a) |
How many area code telephone number combinations were possible under the old rules? (See Exercise 8 for a description of local telephone numbers.) (b) How many area code telephone number combinations are now possible under the new rules? (c) Why do you think it was necessary to make this change? (d) How many area codes that violate Rule (ii) are you personally familiar with? 10.2 Permutations and Combinations LEARNING OBJECTIVES After completing this section, you will be able to: ■ Find the number of permutations ■ Find the number of distinguishable permutations ■ Find the number of combinations ■ Solve counting problems involving both permutations and combinations In this section we single out two important special cases of the Fundamental Counting Principle: permutations and combinations. SE CTI O N 10. 2 | Permutations and Combinations 663 Permutations of three colored squares ■ Permutations A permutation of a set of distinct objects is an ordering of these objects. For example, some permutations of the letters ABCDWXYZ are XAYBZWCD ZAYBCDWX DBWAZXYC YDXAWCZB How many such permutations are possible? Since there are eight choices for the first position, seven for the second (after the first has been chosen), six for the third (after the first two have been chosen), and so on, the Fundamental Counting Principle tells us that the number of possible permutations is 8 7 6 5 4 3 2 1 40,320 This same reasoning with 8 replaced by n leads to the following observation. The number of permutations of n objects is n!. How many permutations consisting of five letters can be made from these same eight letters? Some of these permutations are XYZWC AZDWX AZXYB WDXZB Again, there are eight choices for the first position, seven for the second, six for the third, five for the fourth, and four for the fifth. By the Fundamental Counting Principle the number of such permutations is 8 7 6 5 4 6720 In general, if a set has n elements, then the number of ways of ordering r elements from and is called the number of permutations of n objects taken n, r P the set is denoted by r at a time. 1 2 1 2 P 8, 5 We have just |
shown that 8, 5 6720 will help us find a general formula for. The same reasoning that was used to find. Indeed, there are n objects and r poP P sitions to place them in. Thus, there are n choices for the first position, n 1 choices for the second, n 2 choices for the third, and so on. The last position can be filled in n r 1 ways. By the Fundamental Counting Principle, n, r 1 2 1 2 n This formula can be written more compactly using factorial notation: n! n r 1! 2 PERMUTATIONS OF n OBJECTS TAKEN r AT A TIME The number of permutations of n objects taken r at a time is n, r P 1 2 n! n r 1! 2 664 CHAPTER 10 | Counting and Probability E X AM P L E 1 | Finding the Number of Permutations A club has nine members. In how many ways can a president, vice president, and secretary be chosen from the members of this club? ▼ SO LUTI O N We need the number of ways of selecting three members, in order, for the positions of president, vice president, and secretary from the nine club members. This number is P 9, 3 9! 9 3 9! 6! 2 ✎ Practice what you’ve learned: Do Exercise 23.! 1 1 2 9 8 7 504 ▲ E X AM P L E 2 | Finding the Number of Permutations From 20 raffle tickets in a hat, four tickets are to be selected in order. The holder of the first ticket wins a car, the second a motorcycle, the third a bicycle, and the fourth a skateboard. In how many different ways can these prizes be awarded? ▼ SO LUTI O N The order in which the tickets are chosen determines who wins each prize. So we need to find the number of ways of selecting four objects, in order, from 20 objects (the tickets). This number is P 20, 4 20! 20 4 ✎ Practice what you’ve learned: Do Exercise 33. 20! 16!! 1 1 2 2 20 19 18 17 116,280 ▲ 1 2 P 10, 10 10! ■ Distinguishable Permutations If we have a collection of ten balls, each a different color, then the number |
of permutations. If all ten balls are red, then we have just one distinof these balls is guishable permutation because all the ways of ordering these balls look exactly the same. In general, in considering a set of objects, some of which are of the same kind, then two permutations are distinguishable if one cannot be obtained from the other by interchanging the positions of elements of the same kind. For example, if we have ten balls, of which six are red and the other four are each a different color, then how many distinguishable permutations are possible? The key point here is that balls of the same color are not distinguishable. So each rearrangement of the red balls, keeping all the other balls fixed, gives essentially the same permutation. Since there are 6! rearrangements of the red balls for each fixed position of the other balls, the total number of distinguishable permutations is 10!/6!. The same type of reasoning gives the following general rule. DISTINGUISHABLE PERMUTATIONS If a set of n objects consists of k different kinds of objects with n1 objects of the first kind, n2 objects of the second kind, n3 objects of the third kind, and so on, n2 where n1 these objects is n, then the number of distinguishable permutations of... nk n! n1! n2! n3!... nk! SE CTI O N 10. 2 | Permutations and Combinations 665 E X AM P L E 3 | Finding the Number of Distinguishable Permutations Find the number of different ways of placing 15 balls in a row given that 4 are red, 3 are yellow, 6 are black, and 2 are blue. ▼ SO LUTI O N We want to find the number of distinguishable permutations of these balls. By the formula this number is 15! 4! 3! 6! 2! 6,306,300 ✎ Practice what you’ve learned: Do Exercise 39. ▲ Suppose we have 15 wooden balls in a row and four colors of paint: red, yellow, black, and blue. In how many different ways can the 15 balls be painted in such a way that we have 4 red, 3 yellow, 6 black, and 2 blue balls? A little thought will show that this number is exactly the same as that calculated in Example |
3. This way of looking at the problem is somewhat different, however. Here we think of the number of ways to partition the balls into four groups, each containing 4, 3, 6, and 2 balls to be painted red, yellow, black, and blue, respectively. The next example shows how this reasoning is used. E X AM P L E 4 | Finding the Number of Partitions Fourteen construction workers are to be assigned to three different tasks. Seven workers are needed for mixing cement, five for laying bricks, and two for carrying the bricks to the brick layers. In how many different ways can the workers be assigned to these tasks? ▼ SO LUTI O N We need to partition the workers into three groups containing 7, 5, and 2 workers, respectively. This number is 14! 7! 5! 2! 72,072 ✎ Practice what you’ve learned: Do Exercise 43. ▲ ■ Combinations When finding permutations, we are interested in the number of ways of ordering elements of a set. In many counting problems, however, order is not important. For example, a poker hand is the same hand, regardless of how it is ordered. A poker player who is interested in the number of possible hands wants to know the number of ways of drawing five cards from 52 cards, without regard to the order in which the cards of a given hand are dealt. We now develop a formula for counting in situations such as this, in which order doesn’t matter. A combination of r elements of a set is any subset of r elements from the set (without regard to order). If the set has n elements, then the number of combinations of r elements is deand is called the number of combinations of n elements taken r at a time. noted by For example, consider a set with the four elements, A, B, C, and D. The combinations n, r C 2 1 of these four elements taken three at a time are ABC ABD ACD BCD The permutations of these elements taken three at a time are ABC ACB BAC BCA CAB CBA ABD ADB BAD BDA DAB DBA ACD ADC CAD CDA DAC DCA BCD BDC CBD CDB DBC DCB 666 CHAPTER 10 | Counting and Probability by C n, r 2 but it is customary to use the in the context of In Section 9.6 we denoted n |
, r 2 1 notation counting. For an explanation of why these are the same, see Exercise 82. n, r C 1 2 1 We notice that the number of combinations is a lot fewer than the number of permutations. In fact, each combination of three elements generates 3! permutations. So 4, 3 C 1 2 P 1 4, 3 3! 2 4! 4 3! 2 3! 1 4 In general, each combination of r objects gives rise to r! permutations of these objects. Thus, n, r C 1 2 P 1 n, r r! 2 n! n r! 2 r! 1 COMBINATIONS OF n OBJECTS TAKEN r AT A TIME The number of combinations of n objects taken r at a time is n, r C 1 2 n! n r! 2 r! 1 The key difference between permutations and combinations is order. If we are interested in ordered arrangements, then we are counting permutations; but if we are concerned with subsets without regard to order, then we are counting combinations. Compare Examples 5 and 6 below (where order doesn’t matter) with Examples 1 and 2 (where order does matter). E X AM P L E 5 | Finding the Number of Combinations A club has nine members. In how many ways can a committee of three be chosen from the members of this club? ▼ SO LUTI O N We need the number of ways of choosing three of the nine members. Order is not important here, because the committee is the same no matter how its members are ordered. So we want the number of combinations of nine objects (the club members) taken three at a time. This number is 9, 3 C 1 2 9! 9 3! 2 3! 1 9! 3!6! 9 8 7 3 2 1 84 ✎ Practice what you’ve learned: Do Exercise 47. ▲ Ronald Graham, born in Taft, California, in 1935, is considered the world’s leading mathematician in the field of combinatorics, the branch of mathematics that deals with counting. For many years Graham headed the Mathematical Studies Center at Bell Laboratories in Murray Hill, New Jersey, where he solved key problems for the telephone indus- sion schedules so that the three astronauts aboard the spacecraft could find the time to perform all the necessary tasks. The number of ways to allot these tasks was astronomical—too vast for even a computer to sort |
out. Graham, using his knowledge of combinatorics, was able to reassure NASA that there were easy ways of solving their problem that were not too far from the theoretically best possible solution. Besides being a prolific mathematician, Graham is an accomplished juggler (he has been on stage with the Cirque du Soleil and is a past president of the International Jugglers Association). Several of his research papers address the mathematical aspects of juggling. He is also fluent in Mandarin Chinese and Japanese and once spoke with former President Jiang of China in his native language © try. During the Apollo program, NASA needed to evaluate mis- SE CTI O N 10. 2 | Permutations and Combinations 667 E X AM P L E 6 | Finding the Number of Combinations From 20 raffle tickets in a hat, four tickets are to be chosen at random. The holders of the winning tickets are to be awarded free trips to the Bahamas. In how many ways can the four winners be chosen? ▼ SO LUTI O N We need to find the number of ways of choosing four winners from 20 entries. The order in which the tickets are chosen doesn’t matter, because the same prize is awarded to each of the four winners. So we want the number of combinations of 20 objects (the tickets) taken four at a time. This number is C 20, 4 20! 20 4 ✎ Practice what you’ve learned: Do Exercise 51. 20! 4!16! 4!! 1 2 2 1 20 19 18 17 4 3 2 1 4845 ▲ If a set S has n elements, then is the number of ways of choosing k elements from S, that is, the number of k-element subsets of S. Thus, the number of subsets of S of all possible sizes is given by the sum n, k C 2 1 n, 0 C 1 2 C n, 1 1 2 C n, 2 1 2... C 2n n, n 1 2 (See Section 9.6, Exercise 56, where this sum is discussed.) A set with n elements has 2n subsets. E X AM P L E 7 | Finding the Number of Subsets of a Set A pizza parlor offers the basic cheese pizza and a choice of 16 toppings. How many different kinds of pizza can be ordered at this pizza parlor? ▼ SO LUTI O N We |
need the number of possible subsets of the 16 toppings (including the empty set, which corresponds to a plain cheese pizza). Thus, 216 65,536 different pizzas can be ordered. ✎ Practice what you’ve learned: Do Exercise 61. ▲ ■ Problem Solving with Permutations and Combinations The crucial step in solving counting problems is deciding whether to use permutations, combinations, or the Fundamental Counting Principle. In some cases the solution of a problem may require using more than one of these principles. Here are some general guidelines to help us decide how to apply these principles. GUIDELINES FOR SOLVING COUNTING PROBLEMS 1. Fundamental Counting Principle. When consecutive choices are being made, use the Fundamental Counting Principle. 2. Does the Order Matter? When we want to find the number of ways of pick- ing r objects from n objects, we need to ask ourselves, “Does the order in which we pick the objects matter?” If the order matters, we use permutations. If the order doesn’t matter, we use combinations. 668 CHAPTER 10 | Counting and Probability E X AM P L E 8 | A Problem Involving Combinations A group of 25 campers contains 15 women and 10 men. In how many ways can a scouting party of 5 be chosen if it must consist of 3 women and 2 men? Three women can be chosen from the 15 women in the group in ▼ SO LUTI O N 15, 3 2 1 ways. Thus, ways, and two men can be chosen from the 10 men in the group in by the Fundamental Counting Principle the number of ways of choosing the scouting party is 10, 2 C C 1 2 15, 3 2 ✎ Practice what you’ve learned: Do Exercise 63. 10, 2 C 2 1 1 C 455 45 20,475 ▲ E X AM P L E 9 | A Problem Involving Permutations and Combinations A committee of seven—consisting of a chairman, a vice chairman, a secretary, and four other members—is to be chosen from a class of 20 students. In how many ways can this committee be chosen? ▼ SO LUTI O N of choosing them is In choosing the three officers, order is important. So the number of ways 20, 4 We could have first chosen the four unordered members of the committee—in |
C officers from the remaining 16 members, in gives the same answer. ways—and then the three ways. Check that this 16, 3 P 2 2 1 1 P 20, 3 6840 1 2 Next, we need to choose four other students from the 17 remaining. Since order doesn’t matter in this case, the number of ways of doing this is 2380 2 Thus, by the Fundamental Counting Principle the number of ways of choosing this committee is 17, 4 C 1 C ✎ Practice what you’ve learned: Do Exercise 65. 17, 4 20, 3 P 1 2 2 1 6840 2380 16,279,200 ▲ E X AM P L E 10 | A Group Photograph Twelve employees at a company picnic are to stand in a row for a group photograph. In how many ways can this be done if (a) Jane and John insist on standing next to each other? (b) Jane and John refuse to stand next to each other? Jane John ▼ SO LUTI O N (a) Since the order in which the people stand is important, we use permutations. But we can’t use the formula for permutations directly. Since Jane and John insist on standing together, let’s think of them as one object. Thus, we have 11 objects to arrange in a row, and there are ways of doing this. For each of these arrangements 11, 11 P 1 2 SE CTI O N 10. 2 | Permutations and Combinations 669 there are two ways of having Jane and John stand together: Jane-John or John-Jane. Thus, by the Fundamental Counting Principle the total number of arrangements is 2 P 11, 11 1 2 2 11! 79,833,600 (b) There are P 12, 12 have ways of arranging the 12 people. Of these, Jane and John standing together (by part (a)). All the rest have Jane and John standing apart. So the number of arrangements with Jane and John apart is 11, 11 2 1 2 1 2 P P 12, 12 2 P 11, 11 12! 2 11! 399,168,000 1 ✎ Practice what you’ve learned: Do Exercises 71 and 77. 1 2 2 ▲ 10. ▼ CONCE PTS 1. True or false? In counting combinations, order matters. 2. True or false? In counting permutations, order matters. 3. True or false? For a |
set of n distinct objects, the number of different combinations of these objects is more than the number of different permutations. 4. True or false? If we have a set with five distinct objects, then the number of different ways of choosing two members of this set is the same as the number of ways of choosing three members. ▼ SKI LLS 5–10 ■ Evaluate the expression. 5. 8. 8, 3 2 10, 5 P P 1 1 2 6. 9. 9, 2 2 100, 1 P P 1 1 2 7. 10. 11, 4 99, 3 P P 1 1 2 2 11–16 ■ Find the number of distinguishable permutations of the given letters. 11. AAABBC 13. AABCD 12. AAABBBCCC 14. ABCDDDEE 15. XXYYYZZZZ 16. XXYYZZZ 17–22 ■ Evaluate the expression. 17. C 20. C 8, 3 2 10, 5 1 1 2 18. C 21. C 1 1 9, 2 2 100, 1 2 19. C 22. C 11, 4 99, 3 1 1 2 2 ▼ APPLICATIONS 23–36 ■ These problems involve permutations. ✎ 23. Class Officers In how many different ways can a president, vice president, and secretary be chosen from a class of 15 students? 24. Contest Prizes In how many different ways can first, second, and third prizes be awarded in a game with eight contestants? 25. Seating Arrangements In how many different ways can six of ten people be seated in a row of six chairs? 26. Seating Arrangements In how many different ways can six people be seated in a row of six chairs? 27. Three-Letter Words How many three-letter “words” can be made from the letters FGHIJK? (Letters may not be repeated.) 28. Letter Permutations How many permutations are possible from the letters of the word LOVE? 29. Three-Digit Numbers How many different three-digit whole numbers can be formed by using the digits 1, 3, 5, and 7 if no repetition of digits is allowed? 30. Piano Recital A pianist plans to play eight pieces at a recital. In how many ways can she arrange these pieces in the program? 31. Running a Race In how many different ways can a race with nine runners be completed, |
assuming that there is no tie? 32. Signal Flags A ship carries five signal flags of different colors. How many different signals can be sent by hoisting exactly three of the five flags on the ship’s flagpole in different orders? ✎ 33. Contest Prizes In how many ways can first, second, and third prizes be awarded in a contest with 1000 contestants? 34. Class Officers In how many ways can a president, vice president, secretary, and treasurer be chosen from a class of 30 students? 35. Seating Arrangements In how many ways can five stu- dents be seated in a row of five chairs if Jack insists on sitting in the first chair? Jack 36. Seating Arrangements In how many ways can the students in Exercise 35 be seated if Jack insists on sitting in the middle chair? 670 CHAPTER 10 | Counting and Probability 37–44 ■ These problems involve distinguishable permutations. 37. Arrangements In how many ways can two blue marbles and four red marbles be arranged in a row? 49. Draw Poker Hands How many different five-card hands can be dealt from a deck of 52 cards? 38. Arrangements In how many different ways can five red balls, two white balls, and seven blue balls be arranged in a row? ✎ 39. Arranging Coins In how many different ways can four pen- nies, three nickels, two dimes, and three quarters be arranged in a row? 40. Arranging Letters In how many different ways can the let- ters of the word ELEEMOSYNARY be arranged? 41. Distributions A man bought three vanilla ice-cream cones, two chocolate cones, four strawberry cones, and five butterscotch cones for his 14 chidren. In how many ways can he distribute the cones among his children? 42. Room Assignments When seven students take a trip, they find a hotel with three rooms available: a room for one person, a room for two people, and a room for three people. In how many different ways can the students be assigned to these rooms? (One student has to sleep in the car.) ✎ 43. Work Assignments Eight workers are cleaning a large house. Five are needed |
to clean windows, two to clean the carpets, and one to clean the rest of the house. In how many different ways can these tasks be assigned to the eight workers? 44. Jogging Routes A jogger jogs every morning to his health club, which is eight blocks east and five blocks north of his home. He always takes a route that is as short as possible, but he likes to vary it (see the figure). How many different routes can he take? ENNEEENENEENE, where E is East and N is North.] [Hint: The route shown can be thought of as Health Club Home 45–58 ■ These problems involve combinations. 45. Choosing Books In how many ways can three books be chosen from a group of six different books? 46. Pizza Toppings In how many ways can three pizza toppings be chosen from 12 available toppings? ✎ 47. Committee In how many ways can a committee of three members be chosen from a club of 25 members? 50. Stud Poker Hands How many different seven-card hands can be picked from a deck of 52 cards? ✎ 51. Choosing Exam Questions A student must answer seven of the ten questions on an exam. In how many ways can she choose the seven questions? 52. Three-Topping Pizzas A pizza parlor offers a choice of 16 different toppings. How many three-topping pizzas are possible? 53. Violin Recital A violinist has practiced 12 pieces. In how many ways can he choose eight of these pieces for a recital? 54. Choosing Clothing If a woman has eight skirts, in how many ways can she choose five of these to take on a weekend trip? 55. Field Trip In how many ways can seven students from a class of 30 be chosen for a field trip? 56. Field Trip In how many ways can the seven students in Exer- cise 55 be chosen if Jack must go on the field trip? 57. Field Trip In how many ways can the seven students in Exercise 55 be chosen if Jack is not allowed to go on the field trip? 58. Lottery In the 6/49 lottery game, a player picks six numbers from 1 to 49. How many different choices does the player have? 59. Subsets A set has eight elements. (a) How many subsets containing five elements does |
this set have? (b) How many subsets does this set have? 60. Travel Brochures A travel agency has limited numbers of eight different free brochures about Australia. The agent tells you to take any that you like but no more than one of any kind. In how many different ways can you choose brochures (including not choosing any)? ✎ 61. Hamburgers A hamburger chain gives their customers a choice of ten different hamburger toppings. In how many different ways can a customer order a hamburger? 62. To Shop or Not to Shop Each of 20 shoppers in a shopping mall chooses to enter or not to enter the Dressfastic clothing store. How many different outcomes of their decisions are possible? 63–79 ■ Solve the problem using the appropriate counting principle(s). ✎ 63. Choosing a Committee A class has 20 students, of whom 12 are females and 8 are males. In how many ways can a committee of five students be picked from this class under each condition? (a) No restriction is placed on the number of males or females on the committee. 48. Choosing a Group In how many ways can six people be chosen from a group of ten? (b) No males are to be included on the committee. (c) The committee must have three females and two males. 64. Doubles Tennis From a group of ten male and ten female tennis players, two men and two women are to face each other in a men-versus-women doubles match. In how many different ways can this match be arranged? ✎ ✎ 65. Choosing a Committee A committee of six is to be chosen from a class of 20 students. The committee is to consist of a president, a vice president, and four other members. In how many different ways can the committee be picked? 66. Choosing a Group Sixteen boys and nine girls go on a camping trip. In how many ways can a group of six be selected to gather firewood, given the following conditions? (a) The group consists of two girls and four boys. (b) The group contains at least two girls. 67. Dance Committee A school dance committee is to consist of two freshmen, three sophomores, four juniors, and five seniors. If six freshmen, eight sophomores, twelve juniors, and ten seniors are eligible to be on the committee, in how many ways can the committee be chosen? 68. |
Casting a Play A group of 22 aspiring thespians contains ten men and twelve women. For the next play the director wants to choose a leading man, a leading lady, a supporting male role, a supporting female role, and eight extras—three women and five men. In how many ways can the cast be chosen? 69. Hockey Lineup A hockey team has 20 players, of whom twelve play forward, six play defense, and two are goalies. In how many ways can the coach pick a starting lineup consisting of three forwards, two defense players, and one goalie? 70. Choosing a Pizza A pizza parlor offers four sizes of pizza (small, medium, large, and colossus), two types of crust (thick and thin), and 14 different toppings. How many different pizzas can be made with these choices? ✎ 71. Arranging a Class Picture In how many ways can ten students be arranged in a row for a class picture if John and Jane want to stand next to each other and Mike and Molly also insist on standing next to each other? 72. Arranging a Class Picture In how many ways can the ten students in Exercise 71 be arranged if Mike and Molly insist on standing together but John and Jane refuse to stand next to each other? 73. Seating Arrangements In how many ways can four men and four women be seated in a row of eight seats for each of the following arrangements? (a) The first seat is to be occupied by a man. (b) The first and last seats are to be occupied by women. 74. Seating Arrangements In how many ways can four men and four women be seated in a row of eight seats for each of the following arrangements? (a) The women are to be seated together. (b) The men and women are to be seated alternately by gender. 75. Selecting Prizewinners From a group of 30 contestants, six are to be chosen as semifinalists, then two of those are chosen as finalists, and then the top prize is awarded to one of the finalists. In how many ways can these choices be made in sequence? 76. Choosing a Delegation Three delegates are to be chosen from a group of four lawyers, a priest, and three professors. In SE CTI ON 10. 2 | Permutations and Combinations 671 how |
many ways can the delegation be chosen if it must include at least one professor? 77. Choosing a Committee In how many ways can a committee of four be chosen from a group of ten if two people refuse to serve together on the same committee? 78. Geometry Twelve dots are drawn on a page in such a way that no three are collinear. How many straight lines can be formed by joining the dots? 79. Parking Committee A five-person committee consisting of students and teachers is being formed to study the issue of student parking privileges. Of those who have expressed an interest in serving on the committee, 12 are teachers and 14 are students. In how many ways can the committee be formed if at least one student and one teacher must be included? ▼ DISCOVE RY • DISCUSSION • WRITI NG 80. Complementary Combinations Without performing any calculations, explain in words why the number of ways of choosing two objects from ten objects is the same as the number of ways of choosing eight objects from ten objects. In general, explain why n, r C 1 2 C 1 n, n r 2 81. An Identity Involving Combinations Kevin has ten different marbles, and he wants to give three of them to Luke and two to Mark. In how many ways can he choose to do this? There are two ways of analyzing this problem: He could first pick three for Luke and then two for Mark, or he could first pick two for Mark and then three for Luke. Explain how these two viewpoints show that 2 2 # C 1 1 In general, explain why 10, 3 C 2 7, 2 C 1 10, 2 2 # C 8, 3 1 2 C n, r n r, k # C 1 the Same as 2 # C C n, k n k, r 2 1? This exercise explains 1 2 82. Why Is 1 C n, r 2 n why the binomial coefficients r 2 1 n, r C of 1 choosing r objects from n objects. First, note that expanding a binomial using only the Distributive Property gives that appear in the expansion, the number of ways of are the same as xx xy yx yy 1 y 3 x y 1 2 1 xx xy yx yy 2 xxx xxy xyx xyy yxx yxy yyx yyy (a) Expand (b) Write all the terms that represent x |
2y 3 together. These are using only the Distributive Property. 1 2 x y 5 all the terms that contain two x’s and three y’s. (c) Note that the two x’s appear in all possible positions. Conclude that the number of terms that represent x 2y 3 is C 5, 2. 1 2 (d) In general, explain why C same as n, r. 1 2 in the Binomial Theorem is the n r 2 1 672 CHAPTER 10 | Counting and Probability 10.3 Probability LEARNING OBJECTIVES After completing this section, you will be able to: ■ Find the probability of an event by counting ■ Find the probability of the complement of an event ■ Find the probability of the union of events ■ Find the probability of the intersection of independent events In the preceding chapters we modeled real-world situations using precise rules, such as equations or functions. But many of our everyday activities are not governed by precise rules; rather, they involve randomness and uncertainty. How can we model such situations? How can we find reliable patterns in random events? In this section we will see how the ideas of probability provide answers to these questions. Let’s look at a simple example. We roll a die, and we’re hoping to get a “two.” Of course, it’s impossible to predict what number will show up. But here’s the key idea: If we roll the die many many times, the number two will show up about one-sixth of the time. This is because each of the six numbers, 1, 2, 3, 4, 5, and 6, is equally likely to show up, so the “two” will show up about a sixth of the time. If you try this experiment, you will see that it actually works! We will say that the probability (or chance) of getting 1 a “two” is. 6 If we pick a card from a 52-card deck, what are the chances that it is an ace? Again, each card is equally likely to be picked. Since there are four aces, the probability (or chance) of picking an ace is. 4 52 Probability plays a key role in many of the sciences. A remarkable example of the use of probability is Gregor Mendel’s discovery of genes (which he could not see) by applying probabilistic |
reasoning to the patterns that he saw in inherited traits. Today, probability is an indispensable tool for decision making in business, industry, government, and scientific research. For example, probability is used to determine the effectiveness of new medicines, assess fair prices for insurance policies, and gauge public opinion on a topic (without interviewing everyone). In the remaining sections of this chapter we will see how some of these applications are possible. ■ What Is Probability? To discuss probability, let’s begin by defining some terms. An experiment is a process, such as tossing a coin or rolling a die, that gives definite results, called the outcomes of the experiment. For tossing a coin, the possible outcomes are “heads” and “tails”; for rolling a die, the outcomes are 1, 2, 3, 4, 5, and 6. The sample space of an experiment is the set of all possible outcomes. If we let H stand for heads and T for tails, then the sample space. of the coin-tossing experiment is S H, T 5 The table lists some experiments and the corresponding sample spaces. 6 The mathematical theory of probability was first discussed in 1654 in a series of letters between Pascal (see page 636) and Fermat (see page 159). Their correspondence was prompted by a question raised by the experienced gambler the Chevalier de Méré. The Chevalier was interested in the equitable distribution of the stakes of an interrupted gambling game (see Problem 3, page 700). SECTION 1 0.3 | Probability 673 Experiment Sample space Tossing a coin Rolling a die Tossing a coin twice and observing the sequence of heads and tails Picking a card from a deck and observing the suit Administering a drug to three patients and observing whether they recover (R) or not (N ) 6 H, T 5 1, 2, 3, 4, 5, 6 5 HH, HT, TH, TT 5 6 6 RRR, RRN, RNR, RNN, 5 NRR, NRN, NNR, NNN 6 We will be concerned only with experiments for which all the outcomes are equally likely. We already have an intuitive feeling for what this means. When we toss a perfectly balanced coin, heads and tails are equally likely outcomes in the sense that if this experiment is repeated many times, we expect that about as many heads as tails will |
show up. In any given experiment we are often concerned with a particular set of outcomes. We might be interested in a die showing an even number or in picking an ace from a deck of cards. Any particular set of outcomes is a subset of the sample space. This leads to the following definition. DEFINITION OF AN EVENT If S is the sample space of an experiment, then an event is any subset of the sample space. E X AM P L E 1 | Events in a Sample Space If an experiment consists of tossing a coin three times and recording the results in order, the sample space is S HHH, HHT, HTH, THH, TTH, THT, HTT, TTT 6 5 The event E of showing “exactly two heads” is the subset of S that consists of all outcomes with two heads. Thus E 5 HHT, HTH, THH 6 The event F of showing “at least two heads” is F 5 HHH, HHT, HTH, THH 6 and the event of showing “no heads” is G {TTT}. ✎ Practice what you’ve learned: Do Exercise 5. ▲ We are now ready to define the notion of probability. Intuitively, we know that rolling a die may result in any of six equally likely outcomes, so the chance of any particular outcome occurring is. What is the chance of showing an even number? Of the six equally likely outcomes possible, three are even numbers. So it is reasonable to say that the chance 3. This reasoning is the intuitive basis for the following of showing an even number is 6 definition of probability. 1 2 1 6 674 CHAPTER 10 | Counting and Probability DEFINITION OF PROBABILITY Let S be the sample space of an experiment in which all outcomes are equally likely, and let E be an event. The probability of E, written, is number of elements in E number of elements in S 1 Notice that 0 and 1, that is, so the probability P E 1 2 of an event is a number between 0 P E 1 1 The closer the probability of an event is to 1, the more likely the event is to happen;, then E is called the certain event; and if the closer to 0, the less likely. If P, then E is called the impossible event AM P L E 2 | Finding the Probability of an |
Event A coin is tossed three times, and the results are recorded. What is the probability of getting exactly two heads? At least two heads? No heads? ▼ SO LUTI O N By the results of Example 1 the sample space S of this experiment contains eight outcomes, and the event E of getting “exactly two heads” contains three outcomes, {HHT, HTH, THH}, so by the definition of probability, n 1 n 1 Similarly, the event F of getting “at least two heads” has four outcomes, {HHH, HHT, HTH, THH}, so The event G of getting “no heads” has one element, so ✎ Practice what you’ve learned: Do Exercise 7. ▲ MATHEMATICS IN THE MODERN WORLD Fair Voting Methods The methods of mathematics have recently been applied to problems in the social sciences. For example, how do we find fair voting methods? You may ask, “What is the problem with how we vote in elections?” Well, suppose candidates A, B, and C are running for president. The final vote tally is as follows: A gets 40%, B gets 39%, and C gets 21%. So candidate A wins. But 60% of the voters didn’t want A. Moreover, suppose you voted for C, but you dislike A so much that you would have been willing to change your vote to B to avoid having A win. Most of the voters who voted for C feel the same way you do, so we have a situation in which most of the voters prefer B over A, but A wins. Is that fair? In the 1950s Kenneth Arrow showed mathematically that no democratic method of voting can be completely fair; he later won a Nobel Prize for his work. Mathematicians continue to work on finding fairer voting systems. The system that is most often used in federal, state, and local elections is called plurality voting (the candidate with the most votes wins). Other systems include majority voting (if no candidate gets a majority, a runoff is held between the top two vote-getters), approval voting (each voter can vote for as many candidates as he or she approves of), preference voting (each voter orders the candidates according to his or her preference), and cumulative voting (each voter gets as many votes as there are candidates and can give all of his or |
her votes to one candidate or distribute them among the candidates as he or she sees fit). This last system is often used to select corporate boards of directors. Each system of voting has both advantages and disadvantages. SECTION 1 0.3 | Probability 675 ■ Calculating Probability by Counting To find the probability of an event, we do not need to list all the elements in the sample space and the event. What we do need is the number of elements in these sets. The counting techniques that we learned in the preceding sections will be very useful here. E X AM P L E 3 | Finding the Probability of an Event A five-card poker hand is drawn from a standard deck of 52 cards. What is the probability that all five cards are spades? ▼ SO LUTI O N The experiment here consists of choosing five cards from the deck, and the sample space S consists of all possible five-card hands. Thus, the number of elements in the sample space is n S 1 2 C 1 52, 5 2 52! 52 5! 2 5! 1 2,598,960 The event E that we are interested in consists of choosing five spades. Since the deck contains only 13 spades, the number of ways of choosing five spades is E n 1 2 C 1 13, 5 2 13! 13 5! 2 5! 1 1287 Thus, the probability of drawing five spades is P E 1287 0.0005 2 1 ✎ Practice what you’ve learned: Do Exercise 19. 2,598,960 E S n 1 n 1 2 2 ▲ What does the answer to Example 3 tell us? Since, this means that if you play poker many, many times, on average you will be dealt a hand consisting of only spades about once in every 2000 hands. 0.0005 1 2000 E X AM P L E 4 | Finding the Probability of an Event A bag contains 20 tennis balls, of which four are defective. If two balls are selected at random from the bag, what is the probability that both are defective? ▼ SO LUTI O N The experiment consists of choosing two balls from 20, so the number of elements in the sample space S is. Since there are four defective balls, the num. Thus, the probability of the event E ber of ways of picking two defective balls is |
of picking two defective balls is 20, 2 4 190 ✎ Practice what you’ve learned: Do Exercise 23. 4, 2 1 2 20.032 ▲ The complement of an event E is the set of outcomes in the sample space that is not in E. We denote the complement of an event E by E. We can calculate the probability of E n using the definition and the fact that n E¿ P 1 2 n S 1 E 676 CHAPTER 10 | Counting and Probability PROBABILITY OF THE COMPLEMENT OF AN EVENT Let S be the sample space of an experiment and E an event. Then E¿ P 1 2 1 P E 1 2 This is an extremely useful result, since it is often difficult to calculate the probability can be calculated im- of an event E but easy to find the probability of E, from which P mediately by using this formula. E 2 1 E X AM P L E 5 | Finding the Probability of the Complement of an Event An urn contains 10 red balls and 15 blue balls. Six balls are drawn at random from the urn. What is the probability that at least one ball is red? ▼ SO LUTI O N Let E be the event that at least one red ball is drawn. It is tedious to count all the possible ways in which one or more of the balls drawn are red. So let’s consider E, the complement of this event—namely, that none of the balls that are chosen is red. The number of ways of choosing 6 blue balls from the 15 blue balls is ; the number of ways of choosing 6 balls from the 25 balls is. Thus, 15, 6 25, 6 C C 2 1 E¿ P 1 2 E 15, 6 25, 6 2 2 2 5005 177,100 13 460 By the formula for the complement of an event we have E 1 2, we have 1 P E P 1 2 E¿ 1 2 1 13 460 447 460 0.97 Since E P E¿ P 2 1 P 1 P E FIGURE 1 ✎ Practice what you’ve learned: Do Exercise 25. ▲ ■ The Union of Events If E and F are events, what is the probability that E or F occurs? The word or indicates that we want the probability of the union of these events, that is, E F. So we need to find |
the number of elements in E F. If we simply added the number of elements in E to the number of elements in F, then we would be counting the elements in the overlap twice—once in E and once in F. So to get the correct total, we must subtract the number of elements in E F (see Figure 1). Thus, 1 Using the formula for probability, we get We have proved the following. PROBABILITY OF THE UNION OF TWO EVENTS If E and F are events in a sample space S, then the probability of E or F is SECTION 1 0.3 | Probability 677 E X AM P L E 6 | The Probability of the Union of Events What is the probability that a card drawn at random from a standard 52-card deck is either a face card or a spade? ▼ SO LUTI O N We let E and F denote the following events: E: The card is a face card. F: The card is a spade. Face cards K Q J K Q J K Q J Spades There are 12 face cards and 13 spades in a 52-card deck, so 10 E P 1 2 12 52 and P 13 52 F 1 2 Since 3 cards are simultaneously face cards and spades, we have E F P 1 3 52 2 Thus, by the formula for the probability of the union of two events, we have 11 26 ✎ Practice what you’ve learned: Do Exercise 43(b). 2 13 52 2 1 3 52 1 12 52 2 ▲ ■ The Union of Mutually Exclusive Events Two events that have no outcome in common are said to be mutually exclusive (see Figure 2). For example, in drawing a card from a deck, the events E: The card is an ace. F: The card is a queen. E F FIGURE 2 are mutually exclusive because a card cannot be both an ace and a queen. If E and F are mutually exclusive events, then E F contains no elements. Thus, E F 0, so We have proved the following formula. P PROBABILITY OF THE UNION OF MUTUALLY EXCLUSIVE EVENTS If E and F are mutually exclusive events in a sample space S, then the probability of E or F is There is a natural extension of this formula for any number of mutually exclusive events: If are pairwise mutually exclusive, then E1, E2,..., En E1 E2 P 1... En2 P E12 1 |
P E22 1... P En2 1 678 CHAPTER 10 | Counting and Probability Sevens 7 7 7 7 Face cards AM P L E 7 | The Probability of the Union of Mutually Exclusive Events A card is drawn at random from a standard deck of 52 cards. What is the probability that the card is either a seven or a face card? ▼ SO LUTI O N Let E and F denote the following events. E: The card is a seven. F: The card is a face card. Since a card cannot be both a seven and a face card, the events are mutually exclusive. We want the probability of E or F, in other words, the probability of E F. By the formula, 12 2 52 ✎ Practice what you’ve learned: Do Exercise 43(a). 4 52 13 ▲ ■ The Intersection of Independent Events We have considered the probability of events joined by the word or, that is, the union of events. Now we study the probability of events joined by the word and—in other words, the intersection of events. When the occurrence of one event does not affect the probability of another event, we say that the events are independent. For instance, if a balanced coin is tossed, the proba1 bility of showing heads on the second toss is, regardless of the outcome of the first toss. 2 So any two tosses of a coin are independent. PROBABILITY OF THE INTERSECTION OF INDEPENDENT EVENTS If E and F are independent events in a sample space S, then the probability of E and F is AM P L E 8 | The Probability of Independent Events A jar contains five red balls and four black balls. A ball is drawn at random from the jar and then replaced; then another ball is picked. What is the probability that both balls are red? 5 ▼ SO LUTI O N The events are independent. The probability that the first ball is red is. 9 5 The probability that the second is red is also. Thus, the probability that both balls are red is 9 5 9 5 9 25 81 0.31 ✎ Practice what you’ve learned: Do Exercise 55. ▲ E X AM P L E 9 | The Birthday Problem What is the probability that in a class of 35 students, at least two have the same birthday? ▼ SO LUTI O N It is reasonable to assume that |
the 35 birthdays are independent and that each day of the 365 days in a year is equally likely as a date of birth. (We ignore February 29.) Let E be the event that two of the students have the same birthday. It is tedious to list all the possible ways in which at least two of the students have matching birthdays. So we consider the complementary event E, that is, that no two students have the same birthday. To SECTION 1 0.3 | Probability 679 Number of people in a group Probability that at least two have the same birthday 5 10 15 20 22 23 24 25 30 35 40 50 0.02714 0.11695 0.25290 0.41144 0.47569 0.50730 0.53834 0.56870 0.70631 0.81438 0.89123 0.97037 find this probability, we consider the students one at a time. The probability that the first student has a birthday is 1, the probability that the second has a birthday different from the first is, and, the probability that the third has a birthday different from the first two is so on. Thus 364 365 363 365 E¿ 2 P 1 P 1 # 364 365 1 P # 363 365 E¿ # 362 #... # 331 0.186 365 365 1 0.186 0.814 E So 2 ✎ Practice what you’ve learned: Do Exercise 67. 1 2 1 ▲ Most people are surprised that the probability in Example 9 is so high. For this reason this problem is sometimes called the “birthday paradox.” The table in the margin gives the probability that two people in a group will share the same birthday for groups of various sizes. 10. ▼ CONCE PTS 1. The set of all possible outcomes of an experiment is called the. A subset of the sample space is called an. 2. The sample space for the experiment of tossing two coins is S HH,, one head” is E 5, HH, 5 getting at least one head is P E 1 2, and the event “getting at least 6,. So the probability of 6 n ____. 2 1 1 n ____ ✎ 7. An experiment consists of tossing a coin twice. (a) Find the sample space. (b) Find the probability of getting heads exactly two times. (c) Find the probability |
of getting heads at least one time. (d) Find the probability of getting heads exactly one time. 8. An experiment consists of tossing a coin and rolling a die. (a) Find the sample space. (b) Find the probability of getting heads and an even number. (c) Find the probability of getting heads and a number greater than 4. (d) Find the probability of getting tails and an odd number. 2 9–10 ■ A die is rolled. Find the probability of the given event. 3. If the intersection of two events E and F is empty, then the events are called from a deck, the event E of “getting a heart” and the event F of. So in drawing a card “getting a spade” are. 4. If the occurrence of an event E does not affect the probability of the occurrence of another event F, then the events are called. So in tossing a coin, the event E of “getting heads on the first toss” and the event F of “getting heads on the second toss” are. ▼ SKI LLS 5. An experiment consists of rolling a die. List the elements in the ✎ following sets. (a) The sample space (b) The event “getting an even number” (c) The event “getting a number greater than 4” 9. (a) The number showing is a six. (b) The number showing is an even number. (c) The number showing is greater than five. 10. (a) The number showing is a two or a three. (b) The number showing is an odd number. (c) The number showing is a number divisible by 3. 11–12 ■ A card is drawn randomly from a standard 52-card deck. Find the probability of the given event. 11. (a) The card drawn is a king. (b) The card drawn is a face card. (c) The card drawn is not a face card. 12. (a) The card drawn is a heart. (b) The card drawn is either a heart or a spade. (c) The card drawn is a heart, a diamond, or a spade. 13–14 ■ A ball is drawn randomly from a jar that contains five red balls, two white balls, and one yellow ball. Find the probability of the given event. 6. |
An experiment consists of tossing a coin and drawing a card from a deck. (a) How many elements does the sample space have? (b) List the elements in the event “getting heads and an ace.” (c) List the elements in the event “getting tails and a face card.” (d) List the elements in the event “getting heads and a spade.” 13. (a) A red ball is drawn. (b) The ball drawn is not yellow. (c) A black ball is drawn. 14. (a) Neither a white nor yellow ball is drawn. (b) A red, white, or yellow ball is drawn. (c) The ball that is drawn is not white. 680 CHAPTER 10 | Counting and Probability 15. A drawer contains an unorganized collection of 18 socks. Three pairs are red, two pairs are white, and four pairs are black. (a) If one sock is drawn at random from the drawer, what is the probability that it is red? (b) Once a sock is drawn and discovered to be red, what is the probability of drawing another red sock to make a matching pair? 16. A child’s game has a spinner as shown in the figure. Find the probability of the given event. (a) The spinner stops on an even number. (b) The spinner stops on an odd number or a number greater than 3. 17. A letter is chosen at random from the word EXTRATERRESTRIAL. Find the probability of the given event. (a) The letter T is chosen. (b) The letter chosen is a vowel. (c) The letter chosen is a consonant. 18. A pair of dice is rolled, and the numbers showing are observed. (a) List the sample space of this experiment. (b) Find the probability of getting a sum of 7. (c) Find the probability of getting a sum of 9. (d) Find the probability that the two dice show doubles (the same number). (e) Find the probability that the two dice show different numbers. (f) Find the probability of getting a sum of 9 or higher. 19–22 ■ A poker hand, consisting of five cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains the cards described. ✎ 19. Five hearts 21. Five face cards 20 |
. Five cards of the same suit 22. An ace, king, queen, jack, and 10 of the same suit (royal flush) ✎ 23. Two balls are picked at random from a jar that contains three red and five white balls. Find the probability of the following events. (a) Both balls are red. (b) Both balls are white. 24. Three CDs are picked at random from a collection of 12 CDs of which four are defective. Find the probability of the following events. (a) All three CDs are defective. (b) All three CDs are functioning properly. ✎ 25. A five-card poker hand is drawn from a standard 52-card deck. Find the probability that at least one card is a spade. 26. A five-card poker hand is drawn from a standard 52-card deck. Find the probability that at least one card is a face card. ▼ APPLICATIONS 27. Four Siblings A couple intends to have four children. As- sume that having a boy and having a girl are equally likely events. (a) List the sample space of this experiment. (b) Find the probability that the couple has only boys. (c) Find the probability that the couple has two boys and two girls. (d) Find the probability that the couple has four children of the same sex. (e) Find the probability that the couple has at least two girls. 28. Bridge Hands What is the probability that a 13-card bridge hand consists of all cards from the same suit? 29. Roulette An American roulette wheel has 38 slots; two slots are numbered 0 and 00, and the remaining slots are numbered from 1 to 36. Find the probability that the ball lands in an oddnumbered slot. 30. Making Words A toddler has wooden blocks showing the letters C, E, F, H, N, and R. Find the probability that the child arranges the letters in the indicated order. (a) In the order FRENCH (b) In alphabetical order 31. Lottery In the 6/49 lottery game, a player selects six numbers from 1 to 49. What is the probability of picking the six winning numbers? 32. An Unlikely Event The president of a large company selects six employees to receive a special bonus. He claims that the six employees are chosen randomly from among the 30 employees, of whom 19 are women and 11 are men. What is |
the probability that no woman is chosen? 33. Guessing on a Test An exam has ten true-false questions. A student who has not studied answers all ten questions by just guessing. Find the probability that the student correctly answers the given number of questions. (a) All ten questions (b) Exactly seven questions 34. Quality Control To control the quality of their product, the Bright-Light Company inspects three light bulbs out of each batch of ten bulbs manufactured. If a defective bulb is found, the batch is discarded. Suppose a batch contains two defective bulbs. What is the probability that the batch will be discarded? 35. Monkeys Typing Shakespeare An often-quoted example of an event of extremely low probability is that a monkey types Shakespeare’s entire play Hamlet by randomly striking keys on a typewriter. Assume that the typewriter has 48 keys (including the space bar) and that the monkey is equally likely to hit any key. (a) Find the probability that such a monkey will actually correctly type just the title of the play as his first word. (b) What is the probability that the monkey will type the phrase “To be or not to be” as his first words? 36. Making Words A monkey is trained to arrange wooden blocks in a straight line. He is then given six blocks showing the letters A, E, H, L, M, T. What is the probability that he will arrange them to spell the word HAMLET? 37. Making Words A monkey is trained to arrange wooden blocks in a straight line. She is then given 11 blocks showing the letters A, B, B, I, I, L, O, P, R, T, Y. What is the probability that the monkey will arrange the blocks to spell the word PROBABILITY? 38. Horse Race Eight horses are entered in a race.You randomly predict a particular order for the horses to complete the race. What is the probability that your prediction is correct? 39. Genetics Many genetic traits are controlled by two genes, one dominant and one recessive. In Gregor Mendel’s original experiments with peas, the genes controlling the height of the plant are denoted by T (tall) and t (short). The gene T is dominant, so a plant with the genotype (genetic makeup) TT or Tt is tall, whereas one with genotype tt is short. By a statistical analysis of the offspring in his |
experiments, Mendel concluded that offspring inherit one gene from each parent and that each possible combination of the two genes is equally likely. If each parent has the genotype Tt, then the following chart gives the possible genotypes of the offspring: Parent 2 T t TT Tt Tt tt Parent 1 T t Find the probability that a given offspring of these parents will be (a) tall or (b) short. 40. Genetics Refer to Exercise 39. Make a chart of the possible genotypes of the offspring if one parent has genotype Tt and the other tt. Find the probability that a given offspring will be (a) tall or (b) short. ▼ SKI LLS 41–42 ■ Determine whether the events E and F in the given experiment are mutually exclusive. 41. The experiment consists of selecting a person at random. (a) E: The person is male. F: The person is female. SECTION 1 0.3 | Probability 681 (b) E: The person is tall. F: The person is blond. 42. The experiment consists of choosing at random a student from your class. (a) E: The student is female. F: The student wears glasses. (b) E: The student has long hair. F: The student is male. 43–44 ■ A die is rolled, and the number showing is observed. Determine whether the events E and F are mutually exclusive. Then find the probability of the event E F. ✎ 43. (a) E: The number is even. F: The number is odd. (b) E: The number is even. F: The number is greater than 4. 44. (a) E: The number is greater than 3. F: The number is less than 5. (b) E: The number is divisible by 3. F: The number is less than 3. 45–46 ■ A card is drawn at random from a standard 52-card deck. Determine whether the events E and F are mutually exclusive. Then find the probability of the event E F. 45. (a) E: The card is a face card. F: The card is a spade. (b) E: The card is a heart. F: The card is a spade. 46. (a) E: The card is a club. F: The card is a king. (b) E: The card is |
an ace. F: The card is a spade. 47–48 ■ Refer to the spinner shown in the figure. Find the probability of the given event. 1515 1414 1313 1212 1111 1010 1616 11 22 77 99 88 33 66 44 55 47. (a) The spinner stops on red. (b) The spinner stops on an even number. (c) The spinner stops on red or an even number. 48. (a) The spinner stops on blue. (b) The spinner stops on an odd number. (c) The spinner stops on blue or an odd number. ▼ APPLICATIONS 49. Roulette An American roulette wheel has 38 slots. Two of the slots are numbered 0 and 00, and the rest are numbered from 682 CHAPTER 10 | Counting and Probability 1 to 36. Find the probability that the ball lands in an oddnumbered slot or in a slot with a number higher than 31. 57. (a) Find the probability that both spinners stop on purple. (b) Find the probability that both spinners stop on blue. 50. Making Words A toddler has eight wooden blocks showing the letters A, E, I, G, L, N, T, and R. What is the probability that the child will arrange the letters to spell one of the words TRIANGLE or INTEGRAL? 51. Choosing a Committee A committee of five is chosen randomly from a group of six males and eight females. What is the probability that the committee includes either all males or all females? 52. Lottery In the 6/49 lottery game a player selects six numbers from 1 to 49. What is the probability of selecting at least five of the six winning numbers? 53. Marbles in a Jar A jar contains six red marbles numbered 1 to 6 and ten blue marbles numbered 1 to 10. A marble is drawn at random from the jar. Find the probability that the given event occurs. (a) The marble is red. (b) The marble is odd-numbered. (c) The marble is red or odd-numbered. (d) The marble is blue or even-numbered. ▼ SKI LLS 54. A coin is tossed twice. Let E and F be the following events: E: The first toss shows heads. F: The second toss shows heads. (a) Are the events |
E and F independent? (b) Find the probability of showing heads on both tosses. ✎ 55. A die is rolled twice. Let E and F be the following events: E: The first roll shows a six. F: The second roll shows a six. (a) Are the events E and F independent? (b) Find the probability of showing a six on both rolls. 56–57 ■ Spinners A and B shown in the figure are spun at the same time. Spinner A Spinner B 56. (a) Are the events “spinner A stops on red” and “spinner B stops on yellow” independent? (b) Find the probability that spinner A stops on red and spinner B stops on yellow. 58. A die is rolled twice. What is the probability of showing a one on both rolls? 59. A die is rolled twice. What is the probability of showing a one on the first roll and an even number on the second roll? 60. A card is drawn from a deck and replaced, and then a second card is drawn. (a) What is the probability that both cards are aces? (b) What is the probability that the first is an ace and the second a spade? ▼ APPLICATIONS 61. Roulette A roulette wheel has 38 slots. Two slots are numbered 0 and 00, and the rest are numbered 1 to 36. A player places a bet on a number between 1 and 36 and wins if a ball thrown into the spinning roulette wheel lands in the slot with the same number. Find the probability of winning on two consecutive spins of the roulette wheel. 62. Making Words A researcher claims that she has taught a monkey to spell the word MONKEY using the five wooden letters E, O, K, M, N, Y. If the monkey has not actually learned anything and is merely arranging the blocks randomly, what is the probability that he will spell the word correctly three consecutive times? 63. Snake Eyes What is the probability of rolling “snake eyes” (double ones) three times in a row with a pair of dice? 64. Lottery In the 6/49 lottery game, a player selects six num- bers from 1 to 49 and wins if he or she selects the winning six numbers. What is the probability of winning the lottery two times in a row? |
65. Balls in a Jar Jar A contains three red balls and four white balls. Jar B contains five red balls and two white balls. Which one of the following ways of randomly selecting balls gives the greatest probability of drawing two red balls? (i) Draw two balls from jar B. (ii) Draw one ball from each jar. (iii) Put all the balls in one jar, and then draw two balls. 66. Slot Machine A slot machine has three wheels. Each wheel has 11 positions: a bar and the digits 0, 1, 2,..., 9. When the handle is pulled, the three wheels spin independently before coming to rest. Find the probability that the wheels stop on the following positions. (a) Three bars (b) The same number on each wheel (c) At least one bar ✎ 67. A Birthday Problem Find the probability that in a group of eight students at least two people have the same birthday. 68. A Birthday Problem What is the probability that in a group of six students at least two have birthdays in the same month? 69. Combination Lock A student has locked her locker with a combination lock, showing numbers from 1 to 40, but she has forgotten the three-number combination that opens the lock. To open the lock, she decides to try all possible combinations. If she can try ten different combinations every minute, what is the probability that she will open the lock within one hour? 70. Committee Membership A mathematics department consists of ten men and eight women. Six mathematics faculty SECTION 1 0.3 | Probability 683 members are to be selected at random for the curriculum committee. (a) What is the probability that two women and four men are selected? (b) What is the probability that two or fewer women are selected? (c) What is the probability that more than two women are selected? 71. Class Photo Twenty students are arranged randomly in a row for a class picture. Paul wants to stand next to Phyllis. Find the probability that he gets his wish. 72. Class Photo Eight boys and 12 girls are arranged in a row. What is the probability that all the boys will be standing at one end of the row and all the girls at the other end? ▼ DISCOVE RY • DISCUSSION • WRITI NG 73. The “Second Son” Paradox Mrs. Smith says, “I have two children. The older one is named William.” |
Mrs. Jones replies, “One of my two children is also named William.” For each woman, list the sample space for the genders of her children, and calculate the probability that her other child is also a son. Explain why these two probabilities are different. 74. The “Oldest Son or Daughter” Phenomenon Poll your class to determine how many of your male classmates are the oldest sons in their families and how many of your female classmates are the oldest daughters in their families. You will most likely find that they form a majority of the class. Explain why a randomly selected individual has a high probability of being the oldest son or daughter in his or her family. DISCOVERY PR OJECT SMALL SAMPLES, BIG RESULTS A national poll finds that voter preference for presidential candidates is as follows: Candidate A: 57% Candidate B: 43% In the poll, 1600 adults were surveyed. Since over 100 million voters participate in a national election, it hardly seems possible that surveying only 1600 adults would be of any value. But it is, and it can be proved mathematically that if the sample of 1600 adults is selected at random, then the results are accurate to within 3% more than 95% of the time. Scientists use these methods to determine properties of a big population by testing a small sample. For example, a small sample of fish from a lake is tested to determine the proportion that is diseased, or a small sample of a manufactured product is tested to determine the proportion that is defective. We can get a feeling for how this works through a simple experiment. Put 1000 white beans and 1000 black beans in a bag, and mix them thoroughly. (It takes a lot of time to count out 1000 beans, so get several friends to count out 100 or so each.) Take a small cup, and scoop up a small sample from the beans. 1. Record the proportion of black (or white) beans in the sample. How closely does the proportion in the sample compare with the actual proportion in the bag? 2. Take several samples, and record the proportion of black (or white) beans in each sample. (a) Graph your results. (b) Average your results. How close is your average to 0.5? 3. Try the experiment again but with 500 black beans and 1500 white beans. What proportion of black (or white) beans would you expect in a sample? 4. Have some friends mix black |
and white beans without telling you the number of each. Estimate the proportion of black (or white) beans by taking a few small samples. 4 5. This bean experiment can be simulated on your graphing calculator. Let’s designate the numbers in the interval [0, 0.25] as “black beans” and those in 0.25, 1 as “white beans.” Now use the random number generator in your cal1 culator to randomly pick a sample of numbers between 0 and 1. Try samples of size 100, 400, 1600, and larger. Determine the proportion of these that are “black beans.” What would you expect this proportion to be? Do your results improve when you use a larger sample? (The TI-83 program in the margin takes a sample of 100 random numbers.) 6. Biologists use sampling techniques to estimate fish populations. For example, to estimate the number of trout in a lake, some of the trout are collected, tagged, and released. Later, a sample is taken, and the number of tagged trout in the sample is recorded. The total number of trout in the lake can be estimated from the following proportionality: number of tagged trout in lake number of trout in lake number of tagged trout in sample number of trout in sample Model this process using beans as follows. Start with a bag containing an unknown number of white beans. Remove a handful of the beans, count them, then “tag” them by marking them with a felt tip pen. Return the tagged beans to the bag, and mix the beans thoroughly. Now take a sample from the bag, and use the number of tagged beans in the sample to estimate the total number of beans in the bag. PROGRAM:SAMPLE :100SN :0SC :For(J,1,N) :randSB :(B0.25)+CSC :End :Disp "PROPORTION=" :Disp C/N 684 SE CTI O N 10.4 | Binomial Probability 685 10.4 Binomial Probability LEARNING OBJECTIVE After completing this section, you will be able to: ■ Find binomial probabilities A coin is weighted so that the probability of heads is. What is the probability of getting exactly two heads in five tosses of this coin? Since the tosses are independent, the probability of getting two heads followed by three tails is Heads Tails Tails Heads Tails |
But this is not the only way we can get exactly two heads. The two heads could occur, for example, on the second toss and the last toss. In this case the probability is Tails Tails Heads Heads Tails In fact, the two heads could occur on any two of the five tosses. Thus, there are ways in which this can happen, each with probability 3. It follows that 2 5, 2 C 1 2 exactly 2 heads in 5 tosses P 1 C 5, 2 1 2 2 a 1 3B A 2 2 3B A 2 3 b 3 1 3 b 0.164609 a The probabilities that we have just calculated are examples of binomial probabilities. In general, a binomial experiment is one in which there are two outcomes, which we call “success” and “failure.” In the coin-tossing experiment described above, “success” is getting “heads,” and “failure” is getting “tails.” The following box tells us how to calculate the probabilities associated with binomial experiments when we perform them many times. BINOMIAL PROBABILITIES An experiment has two possible outcomes, S and F (called “success” and “fail q 1 p P ure”), with 1 r successes in n independent trials of the experiment is. The probability of getting exactly p and P F S 1 2 2 r successes in n trials P 1 C n, r 1 2 2 nr p rq The name “binomial probability” is appropriate because C nomial coefficient n r 2 1 (see Exercise 82 on page 671). n, r 1 2 is the same as the bi- 686 CHAPTER 10 | Counting and Probability E X AM P L E 1 | Rolling a Die A fair die is rolled 10 times. Find the probability of each event. (a) Exactly 2 rolls are sixes. (b) At most 1 roll is a six. (c) At least 2 rolls are sixes. 2 1 S P P and 1 6 ▼ SO LUTI O N We interpret “success” as getting a six and “failure” as not getting a six. 5 Thus,. Since each roll of the die is independent from the others, 6 we can use the formula for binomial probability with n 10, (a) 10, 2 ( |
b) The statement “at most 1 roll is a six” means 0 or 1 roll is a six. So exactly 2 are sixes 8 0.29071 p 1 6 q 5 6, and C 1 6B 5 6B at most 1 roll is a six P 1 2 P P 1 0 or 1 roll is a six 2 P 1 10 0 roll is a six 2 1 C 1 10.161506 0.323011 0.484517 1 roll is a six C 10 Meaning of “at most” P(A or B) = P(A) + P(B) Binomial probability Calculator (c) The statement “at least 2 rolls are sixes” means 2 or more rolls are sixes. Instead of adding the probabilities that 2, 3, 4, 5, 6, 7, 8, 9, or 10 are sixes (which is a lot of work), it’s easier to find the probability of the complement of this event. The complement of “2 or more are sixes” is “0 or 1 is a six.” So 2 or more are sixes P 1 2 0 or 1 is a six 1 P 1 1 0.484517 0.515483 2 P(E) = 1 P(E) From part (b) ✎ Practice what you’ve learned: Do Exercise 21. ▲ E X AM P L E 2 | Testing a Drug for Effectiveness A certain viral disease is known to have a mortality rate of 80%. A drug company has developed a drug that it claims is an effective treatment for the disease. In clinical tests, the drug is administered to ten people suffering from the disease; seven of them recover. (a) What is the probability that seven or more of the patients would have recovered without treatment? (b) Does the drug appear to be effective? ▼ SO LUTI O N (a) The probability of dying from the disease is 80%, or 0.8, so the probability of recovery (“success”) is 1 0.8 0.2. We must calculate the probability that 7, 8, 9, or 10 of the patients would recover without treatment. 7 out of 10 recover 8 out of 10 recover P P 1 1 P 9 out of 10 recover 1 10 out of 10 recover 10, 7 10, 8 0.2 0..8 0.8 2 2 0.8 |
3 0.0007864 2 0.0000737 1 0.0000041 10, 9 2 1 10, 10 0.2 2 0.2 2 1 2 0.8 1 10 2 1 0 0.0000001 2 Adding these probabilities, we find 7 or more recover P 1 0.0008643 2 Number of heads Probability.003906 0.031250 0.109375 0.218750 0.273438 0.218750 0.109375 0.031250 0.003906 SE CTI O N 10.4 | Binomial Probability 687 (b) The probability that seven or more patients would have recovered spontaneously is less than 0.001, or less than of 1%. Thus, it is very unlikely that this happened just by chance. We conclude that the drug is most likely effective. 1 10 ✎ Practice what you’ve learned: Do Exercise 35. ▲ It’s often helpful to know the most likely outcome when a binomial experiment is performed repeatedly. For example, suppose we flip a balanced coin eight times? What is the number of heads that is most likely to show up? To find out, we need to find the probability of getting no heads, one head, two heads, and so on. 0 head 1 head P P 1 1 2 heads, 0 2 a 8, 1 2 a 8.003906 0.03125 0.109375 The probabilities for any number of heads (from 0 to 8) are shown in the table. A bar graph of these probabilities is shown in Figure 1. From the graph we see that the event with greatest probability is four heads and four tails. 0.3 0.25 0.2 Probability 0.15 0.1 0.05 0 FIGURE 1 0 1 2 3 5 4 Heads 6 7 8 10. ▼ CONCE PTS 1. A binomial experiment is one in which there are exactly 5. No successes 6. All successes 7. Exactly one success 8. Exactly one failure outcomes. One outcome is called, and the 9. At least four successes 10. At least three successes other is called. 2. If a binomial experiment has probability p of success, then the probability of failure is exactly r successes in n trials of this experiment is. The probability of getting C 1, p 2 1 p 1 2. ▼ SKI LLS 3–14 |
■ Five independent trials of a binomial experiment with probability of success p 0.7 and probability of failure q 0.3 are performed. Find the probability of each event. 3. Exactly two successes 4. Exactly three successes 11. At most one failure 12. At most two failures 13. At least two successes 14. At most three failures ▼ APPLICATIONS 15. Rolling Dice Six dice are rolled. Find the probability that two of them show a four. 16. Archery An archer hits his target 80% of the time. If he shoots seven arrows, what is the probability of each event? (a) He never hits the target. (b) He hits the target each time. (c) He hits the target more than once. 688 CHAPTER 10 | Counting and Probability (d) He hits the target at least five times. (a) What is the probability that exactly two of the jurors have a college degree? (b) What is the probability that three or more of the jurors have a college degree? 25. Defective Light Bulbs The DimBulb Lighting Company manufactures light bulbs for appliances such as ovens and refrigerators. Typically, 0.5% of their bulbs are defective. From a crate with 100 bulbs, three are tested. Find the probability that the given event occurs. (a) All three bulbs are defective. (b) One or more bulbs is defective. 26. Quality Control An assembly line that manufactures fuses for automotive use is checked every hour to ensure the quality of the finished product. Ten fuses are selected randomly, and if any one of the ten is found to be defective, the process is halted and the machines are recalibrated. Suppose that at a certain time 5% of the fuses being produced are actually defective. What is the probability that the assembly line is halted at that hour’s quality check? 27. Sick Leave The probability that a given worker at the Dyno Nutrition will call in sick on a Monday is 0.04. The packaging department has eight workers. What is the probability that two or more packaging workers will call in sick next Monday? 28. Political Surveys In a certain county, 60% of the voters are in favor of an upcoming school bond initiative. If five voters are interviewed at random, what is the probability that exactly three of them will favor the initiative? 29. Pharmaceuticals A drug that is used to prevent motion sickness is found to be |
effective about 75% of the time. Six friends, prone to seasickness, go on a sailing cruise, and all take the drug. Find the probability of each event. (a) None of the friends gets seasick. (b) All of the friends get seasick. (c) Exactly three get seasick. (d) At least two get seasick. 17. Television Ratings According to a ratings survey, 40% of the households in a certain city tune in to the local evening TV news. If ten households are visited at random, what is the probability that four of them will have their television tuned to the local news? 18. Spread of Disease Health authorities estimate that 10% of the raccoons in a certain rural county are carriers of rabies. A dog is bitten by four different raccoons in this county. What is the probability that he was bitten by at least one rabies carrier? 19. Blood Type About 45% of the population of the United States and Canada have Type O blood. (a) If a random sample of ten people is selected, what is the probability that exactly five have Type O blood? (b) What is the probability that at least three of the random sample of ten have Type O blood? 20. Handedness A psychologist needs 12 left-handed subjects for an experiment, and she interviews 15 potential subjects. About 10% of the population is left-handed. (a) What is the probability that exactly 12 of the potential subjects are left-handed? ✎ (b) What is the probability that 12 or more are left-handed? 21. Germination Rates A certain brand of tomato seeds has a 0.75 probability of germinating. To increase the chance that at least one tomato plant per seed hill germinates, a gardener plants four seeds in each hill. (a) What is the probability that at least one seed germinates in a given hill? (b) What is the probability that two or more seeds will germi- nate in a given hill? (c) What is the probability that all four seeds germinate in a given hill? 22. Genders of Children Assume that for any given live human birth, the chances that the child is a boy or a girl are equally likely. (a) What is the probability that in a family of five children a majority are boys? (b) What is the probability that in a family of seven children a majority |
are girls? 23. Genders of Children The ratio of male to female births is in fact not exactly one-to-one. The probability that a newborn turns out to be a male is about 0.52. A family has ten children. (a) What is the probability that all ten children are boys? (b) What is the probability all are girls? (c) What is the probability that five are girls and five are boys? 24. Education Level In a certain county 20% of the population have a college degree. A jury consisting of 12 people is selected at random from this county. 30. Reliability of a Machine A machine that is used in a manufacturing process has four separate components, each of which has a 0.01 probability of failing on any given day. If any component fails, the entire machine breaks down. Find the probability that on a given day the indicated event occurs. (a) The machine breaks down. (b) The machine does not break down. (c) Only one component does not fail. 31. Genetics Huntington’s disease is a hereditary ailment caused by a recessive gene. If both parents carry the gene but do not have the disease, there is a 0.25 probability that an offspring will fall victim to the condition. A newly wed couple find through genetic testing that they both carry the gene (but do not have the disease). If they intend to have four children, find the probability of each event. (a) At least one child gets the disease. (b) At least three of the children get the disease. 32. Selecting Cards Three cards are randomly selected from a standard 52-card deck, one at a time, with each card replaced in the deck before the next one is picked. Find the probability of each event. (a) All three cards are hearts. (b) Exactly two of the cards are spades. (c) None of the cards is a diamond. (d) At least one of the cards is a club. 33. Smokers and Nonsmokers The participants at a mathematics conference are housed dormitory-style, five to a room. Due to an oversight, conference organizers forget to ask whether the participants are smokers. In fact, it turns out that 30% are smokers. Find the probability that Fred, a nonsmoking conference participant, will be housed with (a) Exactly one smoker (b) One or more |
smokers 34. Telephone Marketing A mortgage company advertises its rates by making unsolicited telephone calls to random numbers. About 2% of the calls reach consumers who are interested in the company’s services. A telephone consultant can make 100 calls per evening shift. (a) What is the probability that two or more calls will reach an interested party in one shift? (b) How many calls does a consultant need to make to ensure at least a 0.5 probability of reaching one or more interested parties? [Hint: Use trial and error.] ✎ 35. Effectiveness of a Drug A certain disease has a mortality rate of 60%. A new drug is tested for its effectiveness against this disease. Ten patients are given the drug, and eight of them recover. (a) Find the probability that eight or more of the patients would have recovered without the drug. (b) Does the drug appear to be effective? (Consider the drug effective if the probability in part (a) is 0.05 or less.) 36. Hitting a Target An archer normally hits the target with probability of 0.6. She hires a new coach for a series of special lessons. After the lessons she hits the target in five out of eight attempts. SE CTI O N 10.4 | Binomial Probability 689 (a) Find the probability that she would have hit five or more out of the eight attempts before her lessons with the new coach. (b) Did the new coaching appear to make a difference? (Consider the coaching effective if the probability in part (a) is 0.05 or less.) ▼ DISCOVE RY • DISCUSSION • WRITI NG 37. Most Likely Outcome (Balanced Coin) A balanced coin is tossed nine times, and the number of heads is observed. The bar graph below shows the probabilities of getting any number of heads from 0 to 9. (Compare this graph to the one on page 687.) (a) Find the probabilities of getting exactly one head, exactly two heads, and so on, to confirm the probabilities given by the graph. (b) What is the most likely outcome (the number of heads with the greatest probability of occurring)? (c) If the coin is tossed 101 times, what number of heads has the greatest probability of occurring? What if the coin is tossed 100 times? 0.3 0.25 0.2 Probability 0.15 0.1 0. |
05 Heads 38. Most Likely Outcome (Unbalanced Coin) An unbalanced coin has a 0.7 probability of showing heads. The coin is tossed nine times, and the number of heads is observed. The bar graph shows the probabilities of getting any number of heads from 0 to 9. (a) Find the probabilities of getting exactly one head, exactly two heads, and so on, to confirm the probabilities given by the graph. (b) What is the most likely outcome(s) (the number of heads with the greatest probability of occurring)? Compare your results to those in Exercise 37(b). 0.3 0.25 0.2 Probability 0.15 0.1 0.05 Heads 690 CHAPTER 10 | Counting and Probability 10.5 Expected Value LEARNING OBJECTIVE After completing this section, you will be able to: ■ Find the expected value of a game FIGURE 1 0.7 In the game shown in Figure 1, you pay $1 to spin the arrow. If the arrow stops in a red region, you get $3 (the dollar you paid plus $2); otherwise, you lose the dollar you paid. If you play this game many times, how much would you expect to win? Or lose? To answer these questions, let’s consider the probabilities of winning and losing. Since three of the 3, and the probability of losing is regions are red, the probability of winning is 10 7. Remember, this means that if you play this game many times, you expect to win 10 “on average” three out of ten times. Suppose you play the game 1000 times. Then you would expect to win 300 times and lose 700 times. Since we win $2 or lose $1 in each game, our expected payoff in 1000 games is 1 2 1 100. In other words, we expect to lose, So the average expected return per game is 1000 on average, 10 cents per game. Another way to view this average is to divide each side of the preceding equation by 1000. Writing E for the result, we get 700 2 0.1 100 0.3 300 2 1 1 2 300 2 1 2 E 1 1 1000 700 2 2 1 2 300 1000 b a 1 1 700 1000 2 2 0.3 1 2 1 1 0.7 2 2 1 Thus, the expected return, or expected value, per game is E a1p1 a2 p2 where a1 is the payoff |
that occurs with probability p1 and a2 is the payoff that occurs with probability p2. This example leads us to the following definition of expected value. DEFINITION OF EXPECTED VALUE A game gives payoffs a1, a2,..., an with probabilities p1, p2,..., pn. The expected value (or expectation) E of this game is E a1p1 a2p2... anpn The expected value is an average expectation per game if the game is played many times. In general, E need not be one of the possible payoffs. In the preceding example the expected value is 10 cents, but notice that it’s impossible to lose exactly 10 cents in any given game. E X AM P L E 1 | Finding an Expected Value A die is rolled, and you receive $1 for each point that shows. What is your expectation? ▼ SO LUTI O N Each face of the die has probability of showing. So you get $1 with probability, $2 with probability, $3 with probability, and so on. Thus, the expected value is 1 6 1 6 1 6 1 6 SE CTION 1 0.5 | Expected Value 691 21 6 a 3.5 This means that if you play this game many times, you will make, on average, $3.50 per game. ✎ Practice what you’ve learned: Do Exercise 3. ▲ E X AM P L E 2 | Finding an Expected Value In Monte Carlo the game of roulette is played on a wheel with slots numbered 0, 1, 2,..., 36. The wheel is spun, and a ball dropped in the wheel is equally likely to end up in any one of the slots. To play the game, you bet $1 on any number other than zero. (For example, you may bet $1 on number 23.) If the ball stops in your slot, you get $36 (the $1 you bet plus $35). Find the expected value of this game. ▼ SO LUTI O N You gain $35 with probability Thus 1 37, and you lose $1 with probability 36 37. E 1 37 2 35 1 1 1 36 37 2 0.027 In other words, if you play this game many times, you would expect to lose 2.7 cents on every dollar you bet (on average). Consequently, the house |
expects to gain 2.7 cents on every dollar that is bet. This expected value is what makes gambling very profitable for the gaming house and very unprofitable for the gambler. ✎ Practice what you’ve learned: Do Exercise 13. ▲ 10. ▼ CONCE PTS 1. If a game gives payoffs of $10 and $100 with probabilities 0.9 and 0.1, respectively, then the expected value of this game is E 0.9 0.1. 2. If you played the game in Exercise 1 many times, then you would expect your average payoff per game to be about $. ▼ SKI LLS 3–12 ■ Find the expected value (or expectation) of the games described. ✎ 3. Mike wins $2 if a coin toss shows heads and $1 if it shows tails. 4. Jane wins $10 if a die roll shows a six, and she loses $1 otherwise. 5. The game consists of drawing a card from a deck. You win $100 if you draw the ace of spades or lose $1 if you draw any other card. 6. Tim wins $3 if a coin toss shows heads or $2 if it shows tails. 7. Carol wins $3 if a die roll shows a six, and she wins $0.50 otherwise. 8. A coin is tossed twice. Albert wins $2 for each heads and must pay $1 for each tails. 9. A die is rolled. Tom wins $2 if the die shows an even number, and he pays $2 otherwise. 10. A card is drawn from a deck. You win $104 if the card is an ace, $26 if it is a face card, and $13 if it is the 8 of clubs. 11. A bag contains two silver dollars and eight slugs. You pay 50 cents to reach into the bag and take a coin, which you get to keep. 12. A bag contains eight white balls and two black balls. John picks two balls at random from the bag, and he wins $5 if he does not pick a black ball. ✎ ▼ APPLICATIONS 13. Roulette In the game of roulette as played in Las Vegas, the wheel has 38 slots. Two slots are numbered 0 and 00, and the rest are numbered 1 to 36. A $1 bet on any number other than 0 or 00 wins $36 ($35 plus the $1 bet). Find |
the expected value of this game. 14. Sweepstakes A sweepstakes offers a first prize of $1,000,000, second prize of $100,000, and third prize of $10,000. Suppose that two million people enter the contest and three names are drawn randomly for the three prizes. 692 CHAPTER 10 | Counting and Probability (a) Find the expected winnings for a person participating in 20. A Game of Chance A bag contains two silver dollars and six this contest. (b) Is it worth paying a dollar to enter this sweepstakes? 15. A Game of Chance A box contains 100 envelopes. Ten en- velopes contain $10 each, ten contain $5 each, two are “unlucky,” and the rest are empty. A player draws an envelope from the box and keeps whatever is in it. If a person draws an unlucky envelope, however, he must pay $100. What is the expectation of a person playing this game? 16. Combination Lock A safe containing $1,000,000 is locked with a combination lock. You pay $1 for one guess at the sixdigit combination. If you open the lock, you get to keep the million dollars. What is your expectation? 17. Gambling on Stocks An investor buys 1000 shares of a risky stock for $5 a share. She estimates that the probability that the stock will rise in value to $20 a share is 0.1 and the probability that it will fall to $1 a share is 0.9. If the only criterion for her decision to buy this stock was the expected value of her profit, did she make a wise investment? 18. Slot Machine A slot machine has three wheels, and each wheel has 11 positions: the digits 0, 1, 2,..., 9 and the picture of a watermelon. When a quarter is placed in the machine and the handle is pulled, the three wheels spin independently and come to rest. When three watermelons show, the payout is $5; otherwise, nothing is paid. What is the expected value of this game? 19. Lottery In a 6/49 lottery game, a player pays $1 and selects six numbers from 1 to 49. Any player who has chosen the six winning numbers wins $1,000,000. Assuming that this is the only way to win, what is the expected value of this game? CHAPTER 10 | RE |
VIEW slugs. A game consists of reaching into the bag and drawing a coin, which you get to keep. Determine the “fair price” of playing this game, that is, the price at which the player can be expected to break even if he or she plays the game many times (in other words, the price at which the player’s expectation is zero). 21. A Game of Chance A game consists of drawing a card from a deck. You win $13 if you draw an ace. What is a “fair price” to pay to play this game? (See Exercise 20.) 22. Lightning Insurance An insurance company has deter- mined that in a certain region the probability of lightning striking a house in a given year is about 0.0003, and the average cost of repairs of lightning damage is $7500 per incident. The company charges $25 per year for lightning insurance. (a) What is the company’s expected value for the net income from each lightning insurance policy? (b) If the company has 450,000 lightning damage policies, what is the company’s expected yearly income from lightning insurance? ▼ DISCOVE RY • DISCUSSION • WRITI NG 23. The Expected Value of a Sweepstakes Contest A magazine clearinghouse holds a sweepstakes contest to sell subscriptions. If you return the winning number, you win $1,000,000. You have a 1-in-20-million chance of winning, but your only cost to enter the contest is a first-class stamp to mail the entry. Use the current price of a first-class stamp to calculate your expected net winnings if you enter this contest. Is it worth entering the sweepstakes? ▼ P R O P E RTI LAS Fundamental Counting Principle (p. 658) E1, E2,..., Ek are events that occur in order and if event If i 1, 2,..., k ni occur in ways 1 n1 occur in order in, p nk then the sequence of events can ways. Ei can n2 2 Permutations (p. 663) A permutation of a set of objects is an ordering of these objects. If the set has n objects, then there are n! permutations of the objects. If a set has n objects, then the number of ways of ordering the r-element subsets of |
the set is denoted and is called the number of permutations of n objects taken r at a time: n, r P 1 2 n, r P 1 2 n! n r 1! 2 Distinguishable Permutations (p. 664) Suppose that a set has n objects of k kinds (where the objects in each kind cannot be distinguished from each other), and suppose that there are n1 objects of the first kind, n2 of the second kind, and so on (so ). Two permutations of the set are distinguishable from each other if one cannot be obtained from the p nk n2 n n1 other simply by interchanging the positions of elements of the same kind. (In other words, the permutations “look” different.) The number of distinguishable permutations of these objects is n! n1!n2! p nk! Combinations (p. 666) A combination of r objects from a set is any subset of the set that contains r elements (without regard to order). If a set has n objects, then the number of combinations of r elements from the set is denoted C combinations of n objects taken r at a time: and is called the number of n, r 11 22 n, r C 1 2 n! n r! 2 r! 1 Permutations or Combinations? (p. 667) When solving a problem that involves counting the number of ways of picking r objects from a set of n objects, we ask, “Does the order in which the objects are picked make a difference?” If the order matters, use permutations. If the order doesn’t matter, use combinations. Sample Spaces and Events (p. 673) An experiment is a process that gives definite results, called the outcomes. (For example, rolling a die results in the outcomes 1, 2, 3, 4, 5, or 6.) The sample space of an experiment is the set of all possible outcomes. An event is any subset of the sample space. (For example, in rolling a die, the event “get an even number” is the subset 2, 4, 6.) 5 6 Probability (p. 674) Suppose that S is the sample space of an experiment in which all outcomes are equally likely and that E is an event in this experiment. The probability of E, denoted is E P, 2 1 number of outcomes in E number |
of outcomes in The probability of any event E satisfies 0 P E 1 2 then E is impossible (will never happen). If 1 then E is certain (will definitely happen). 0, E 2 1, If P P E 1 1 2 The Complement of an Event (p. 676) If S is the sample space of an experiment and E is an event, then the E¿ complement of E (denoted are not in E. The probability of ) is the set of all outcomes in S that E¿ is given by P E¿ 1 P E 1 The Union of Events (pp. 676–677) Suppose E and F are events in a sample space S. 1 2 2 ▼ CO N C E P T S U M MARY Section 10.1 ■ Use the Fundamental Counting Principle Section 10.2 ■ Find the number of permutations ■ Find the number of distinguishable permutations ■ Find the number of combinations ■ Solve counting problems involving both permutations and combinations Section 10.3 ■ Find the probability of an event using counting principles ■ Find the probability of the complement of an event ■ Find the probability of the union of events ■ Find the probability of the intersection of independent events Section 10.4 ■ Find binomial probabilities Section 10.5 ■ Find the expected value of a game CHAPTER 10 | Review 693 The union of E and F is the set of all outcomes in S that are in E F. either E or F (or both). The union of E and F is denoted For any events E and F the probability of their union is P E F E F 1 2 E F The events E and F are mutually exclusive if tually exclusive events E and F the probability of their union is. For mu The Intersection of Events (p. 678) Suppose E and F are events in a sample space S. The intersection of E and F is the set of all outcomes in S that are in both E and F. The intersection of E and F is denoted E F. The events E and F are independent if the occurrence of one of them does not affect the probability of the occurrence of the other. For independent events E and F the probability of their intersection is Binomial Probabilities (p. 685) A binomial experiment is one that has two possible outcomes, S and F (“success” and “failure”). If 1 p, of the experiment |
is S 1 then the probability of getting exactly r successes in n trials q p and F P P 2 2 1 P r successes in n trials C n, r 1 2 2 prqnr 1 Expected Value (p. 690) If a game gives payoffs p1, p2,..., pn, game is a1, a2,..., an with probabilities then the expected value (or expectation) E of this E a1 p1 a2 p2 p an pn Review Exercises 1, 2(a), 6, 9, 12, 15, 18(a) Review Exercises 2(b), 3(b), 10, 17(a) & (b) 16, 19, 20 3(a), 4, 5, 7, 8, 11, 13, 14, 21(a) 17(c), 21(b)–(f), 22 Review Exercises 18(b), 23, 24(a) & (b), 25(d), 26, 28(a) & (d), 29–33, 35–39, 40(a), 41 24(c), 34, 40(b) 23(d), 25(c), 28(b) & (c) 27, 29 Review Exercises 25, 40 Review Exercises 42–44 694 CHAPTER 10 | Counting and Probability ▼ E X E RC I S E S 1. A coin is tossed, a die is rolled, and a card is drawn from a deck. How many possible outcomes does this experiment have? 2. How many three-digit numbers can be formed by using the digits 1, 2, 3, 4, 5, 6 if repetition of digits (a) is allowed? (b) is not allowed? 3. (a) How many different two-element subsets does the set A, E, I, O, U have? 6 5 (b) How many different two-letter “words” can be made by using the letters from the set in part (a)? 4. An airline company overbooks a particular flight and seven passengers are “bumped” from the flight. If 120 passengers are booked on this flight, in how many ways can the airline choose the seven passengers to be bumped? 5. A quiz has ten true-false questions. In how many different ways can a student earn a score of |
exactly 70% on this quiz? 6. A test has ten true-false questions and five multiple-choice questions with four choices for each. In how many ways can this test be completed? 7. If you must answer only eight of ten questions on a test, how many ways do you have of choosing the questions you will omit? 8. An ice-cream store offers 15 flavors of ice cream. The specialty is a banana split with four scoops of ice cream. If each scoop must be a different flavor, how many different banana splits may be ordered? 9. A company uses a different three-letter security code for each of its employees. What is the maximum number of codes this security system can generate? 10. A group of students determines that they can stand in a row for their class picture in 120 different ways. How many students are in this class? 11. A coin is tossed ten times. In how many different ways can the result be three heads and seven tails? 12. The Yukon Territory in Canada uses a license-plate system for automobiles that consists of two letters followed by three numbers. Explain how we can know that fewer than 700,000 autos are licensed in the Yukon. 13. A group of friends have reserved a tennis court. They find that there are ten different ways in which two of them can play a singles game on this court. How many friends are in this group? 14. A pizza parlor advertises that they prepare 2048 different types of pizza. How many toppings does this parlor offer? 15. In Morse code, each letter is represented by a sequence of dots and dashes, with repetition allowed. How many letters can be represented by using Morse code if three or fewer symbols are used? 16. The genetic code is based on the four nucleotides adenine (A), cytosine (C), guanine (G), and thymine (T). These are connected in long strings to form DNA molecules. For example, a sequence in the DNA may look like CAGTGGTACC.... The code uses “words,” all the same length, that are composed of the nucleotides A, C, G, and T. It is known that at least 20 different words exist. What is the minimum word length necessary to generate 20 words? 17. Given 16 subjects from which to choose, in how many ways can a student select � |
�elds of study as follows? (a) A major and a minor (b) A major, a first minor, and a second minor (c) A major and two minors 18. (a) How many three-digit numbers can be formed by using the digits 0, 1,..., 9? (Remember, a three-digit number cannot have 0 as the leftmost digit.) (b) If a number is chosen randomly from the set 0, 1, 2,...,, what is the probability that the number chosen is a 5 1000 three-digit number? 6 19–20 ■ An anagram of a word is a permutation of the letters of that word. For example, anagrams of the word triangle include griantle, integral, and tenalgir. 19. How many anagrams of the word TRIANGLE are possible? 20. How many anagrams are possible from the word MISSISSIPPI? 21. A committee of seven is to be chosen from a group of ten men and eight women. In how many ways can the committee be chosen using each of the following selection requirements? (a) No restriction is placed on the number of men and women on the committee. (b) The committee must have exactly four men and three women. (c) Susie refuses to serve on the committee. (d) At least five women must serve on the committee. (e) At most two men can serve on the committee. (f) The committee is to have a chairman, a vice chairman, a secretary, and four other members. 22. The U.S. Senate has two senators from each of the 50 states. In how many ways can a committee of five senators be chosen if no state is to have two members on the committee? 23. A jar contains ten red balls labeled 0, 1, 2,..., 9 and five white balls labeled 0, 1, 2, 3, 4. If a ball is drawn from the jar, find the probability of the given event. (a) The ball is red. (b) The ball is even-numbered. (c) The ball is white and odd-numbered. (d) The ball is red or odd-numbered. 24. If two balls are drawn from the jar in Exercise 23, find the probability of the given event. (a) Both |
balls are red. (b) One ball is white and the other is red. (c) At least one ball is red. (d) Both balls are red and even-numbered. (e) Both balls are white and odd-numbered. 25. A coin is tossed three times in a row, and the outcomes of each toss are observed. (a) Find the sample space for this experiment. (b) Find the probability of getting three heads. (c) Find the probability of getting two or more heads. (d) Find the probability of getting tails on the first toss. 26. A shelf has ten books: two mysteries, four romance novels, and four mathematics textbooks. If you select a book at random to take to the beach, what is the probability that it turns out to be a mathematics text? 27. A die is rolled, and a card is selected from a standard 52-card deck. What is the probability that both the die and the card show a six? 28. Find the probability that the indicated card is drawn at random from a 52-card deck. (a) An ace (b) An ace or a jack (c) An ace or a spade (d) A red ace 29. A card is drawn from a 52-card deck, a die is rolled, and a coin is tossed. Find the probability of each outcome. (a) The ace of spades, a six, and heads (b) A spade, a six, and heads (c) A face card, a number greater than 3, and heads 30. Two dice are rolled. Find the probability of each outcome. (a) The dice show the same number. (b) The dice show different numbers. 31. Four cards are dealt from a standard 52-card deck. Find the probability that the cards are (a) all kings (b) all spades (c) all the same color 32. In the “numbers game” lottery a player picks a three-digit number (from 000 to 999), and if the number is selected in the drawing, the player wins $500. If another number with the same digits (in any order) is drawn, the player wins $50. John plays the number 159. (a) What is the probability that he will win $500? (b) What is the probability that he will win $50? 33. In a TV game show, a contestant is given � |
�ve cards with a different digit on each and is asked to arrange them to match the price of a brand-new car. If she gets the price right, she wins the car. What is the probability that she wins, assuming that she knows the first digit but must guess the remaining four? 34. A pizza parlor offers 12 different toppings, one of which is anchovies. If a pizza is ordered at random, what is the probability that anchovies is one of the toppings? 35. A drawer contains an unorganized collection of 50 socks; 20 are red and 30 are blue. Suppose the lights go out so that Kathy can’t distinguish the color of the socks. (a) What is the minimum number of socks Kathy must take out of the drawer to be sure of getting a matching pair? (b) If two socks are taken at random from the drawer, what is the probability that they make a matching pair? CHAPTER 10 | Review 695 36. A volleyball team has nine players. In how many ways can a starting lineup be chosen if it consists of two forward players and three defense players? 37. Zip codes consist of five digits. (a) How many different zip codes are possible? (b) How many different zip codes can be read when the envelope is turned upside down? (An upside-down 9 is a 6; and 0, 1, and 8 are the same when read upside down.) (c) What is the probability that a randomly chosen zip code can be read upside down? (d) How many zip codes read the same upside down as right side up? 38. In the Zip4 postal code system, zip codes consist of nine digits. (a) How many different Zip4 codes are possible? (b) How many different Zip4 codes are palindromes? (A palindrome is a number that reads the same from left to right as right to left.) (c) What is the probability that a randomly chosen Zip4 code is a palindrome? 39. Let N 3,600,000. (Note that N 273255.) (a) How many divisors does N have? (b) How many even divisors does N have? (c) How many divisors of N are multiples of 6? (d) What is the probability that a randomly chosen divisor of N is even? 40. A fair die is rolled eight times. Find the |
probability of each event. (a) A six occurs four times. (b) An even number occurs two or more times. 41. Pacific chinook salmon occur in two varieties: white-fleshed and red-fleshed. It is impossible to tell without cutting the fish open whether it is the white or red variety. About 30% of chinooks have white flesh. An angler catches five chinooks. Find the probability of each event. (a) All are white. (b) All are red. (c) Exactly two are white. (d) Three or more are red. 42. Two dice are rolled. John gets $5 if they show the same number, or he pays $1 if they show different numbers. What is the expected value of this game? 43. Three dice are rolled. John gets $5 if they all show the same number; he pays $1 otherwise. What is the expected value of this game? 44. Mary will win $1,000,000 if she can name the 13 original states in the order in which they ratified the U.S. Constitution. Mary has no knowledge of this order, so she makes a guess. What is her expectation? ■ CHAPTER 10 | TEST 1. Alice and Bill have four grandchildren, and they have three framed pictures of each grand- child. They wish to choose one picture of each grandchild to display on the piano in their living room, arranged from oldest to youngest. In how many ways can they do this? 2. A hospital cafeteria offers a fixed-price lunch consisting of a main course, a dessert, and a drink. If there are four main courses, three desserts, and six drinks to pick from, in how many ways can a customer select a meal consisting of one choice from each category? 3. An Internet service provider requires its customers to select a password consisting of four letters followed by three digits. Find how many such passwords are possible in each of the following cases: (a) Repetition of letters and digits is allowed. (b) Repetition of letters and digits is not allowed. 4. Over the past year, John has purchased 30 books. (a) In how many ways can he pick four of these books and arrange them, in order, on his nightstand bookshelf? (b) In how many ways can he choose four of |
these books to take with him on his vacation at the shore? 5. A commuter must travel from Ajax to Barrie and back every day. Four roads join the two cities. The commuter likes to vary the trip as much as possible, so she always leaves and returns by different roads. In how many different ways can she make the round-trip? 6. A pizza parlor offers four sizes of pizza and 14 different toppings. A customer may choose any number of toppings (or no topping at all). How many different pizzas does this parlor offer? 7. An anagram of a word is a rearrangement of the letters of the word. (a) How many anagrams of the word LOVE are possible? (b) How many different anagrams of the word KISSES are possible? 8. A board of directors consisting of eight members is to be chosen from a pool of 30 candidates. The board is to have a chairman, a treasurer, a secretary, and five other members. In how many ways can the board of directors be chosen? 9. One card is drawn from a deck. Find the probability of each event. (a) The card is red. (b) The card is a king. (c) The card is a red king. 10. A jar contains five red balls, numbered 1 to 5, and eight white balls, numbered 1 to 8. A ball is chosen at random from the jar. Find the probability of each event. (a) The ball is red. (b) The ball is even-numbered. (c) The ball is red or even-numbered. 11. Three people are chosen at random from a group of five men and ten women. What is the probability that all three are men? 12. Two dice are rolled. What is the probability of getting doubles? 13. In a group of four students, what is the probability that at least two have the same astrological sign? 14. An unbalanced coin is weighted so that the probability of heads is 0.55. The coin is tossed ten times. (a) What is the probability of getting exactly 6 heads? (b) What is the probability of getting less than 3 heads? 15. You are to draw one card from a deck. If it is an ace, you win $10; if it is a face card, you win $1; otherwise, you lose $0.50. What is |
the expected value of this game? 696 ● CUMUL ATIVE REVIE W TEST | CHAPTERS 9 and 10 1. For each sequence, find the 7th term and the 20th term. (a) 1 3, 2 (b) an 9, 5 7, 4 5, 3 11,... 2n2 1 n3 n 4 (c) The arithmetic sequence with initial term (d) The geometric sequence with initial term a 12 and common ratio (e) The sequence defined recursively by a1 0.01 and an 2an1. a 1 2 and common difference d 3. r 5 6. 2. Calculate the sum. 1 6 4 5 5 3 9 27 81 p 310 8 5 7 5 (b) (a) 3 5 p 19 5 4 9 5 a 2n n0 6 2 2 3 (c) (d) 2 9 2 27 2 81 p 3. Mary and Kevin buy a vacation home for $350,000. They pay $35,000 down and take out a 15-year mortgage for the remainder. If their annual interest rate is 6%, how much will their monthly mortgage payment be? 4. A sequence is defined inductively by a1 induction to prove that an n2. 1 and an an1 2n 1. Use mathematical 5. (a) Use the Binomial Theorem to expand the expression (b) Find the term containing x4 in the binomial expansion of A 2x 1 5. 2B 2x 1 2B A 12. 6. When students receive their e-mail accounts at Oldenburg University they are assigned a randomly selected password, which consists of three letters followed by four digits (for example, ABC1234). (a) How many such passwords are possible? (b) How many passwords consist of three different letters followed by four different digits? (c) The system administrator decides that in the interest of security, no two passwords can contain the same set of letters and digits (regardless of the order), and no character can be repeated in a password. What is the maximum number of users the system can accommodate under these rules? 7. Toftree is a game in which players roll three dice and receive points based on the outcome. Find the probability of each of the following outcomes. (a) All three dice show the same number. (b) All three dice show an even number. (c |
) The sum of the numbers showing is 15. 8. An alumni association holds a “Vegas night” at its annual homecoming event. At one booth, participants play the following dice game: The player pays a fee of $5, rolls a pair of dice, and then gets back $15 if both dice show the same number, or $7 if the dice show numbers that differ by one (such as 2 and 3, or 5 and 4). What is the expected value of this game? 9. A weighted coin has probability p of showing heads and q 1 p of showing tails when tossed. (a) Find the binomial expansion of 1 p q 2 5. If this coin is tossed five times in a row, what event has the probability represented by the term in this binomial expansion that contains p3? (b) If the probability of heads is, find the probability that in five tosses of the coin there are 2 3 2 heads and 3 tails. 697 698 CUMUL ATI VE REVI E W TE S T | Chapters 9 and 10 10. An insect species has white wings which when closed cover the insect’s back, like the wings of a ladybug. Some individuals have black spots on their wings, arranged randomly, with a total of one to five spots. The probability that a randomly selected insect has n spots is 1 4B n 1, 2, 3, 4, or 5 n A. 1 2 (a) What event has probability 5 a n1 A n? Calculate this sum. 1 4B (b) What is the probability that a randomly selected insect has no spots THE MONTE CARLO METHOD A good way to familiarize ourselves with a fact is to experiment with it. For instance, to convince ourselves that the earth is a sphere (which was considered a major paradox at one time), we could go up in a space shuttle to see that it is so; to see whether a given equation is an identity, we might try some special cases to make sure there are no obvious counterexamples. In problems involving probability, we can perform an experiment many times and use the results to estimate the probability in question. In fact, we often model the experiment on a computer, thereby making it feasible to perform the experiment a large number of times. This technique is called the Monte Carlo method, named after the famous gambling casino in Monaco. E X AM |
P L E 1 | The Contestant’s Dilemma In a TV game show, a contestant chooses one of three doors. Behind one of them is a valuable prize; the other two doors have nothing behind them. After the contestant has made her choice, the host opens one of the other two doors—one that he knows does not conceal a prize—and then gives her the opportunity to change her choice. Should the contestant switch or stay, or does it matter? In other words, by switching doors, does she increase, decrease, or leave unchanged her probability of winning? At first, it may seem that switching doors doesn’t make any difference. After all, two doors are left—one with the prize and one without—so it seems reasonable that the contestant has an equal chance of winning or losing. But if you play this game many times, you will find that by switching doors, you actually win about of the time. 2 3 The authors modeled this game on a computer and found that in one million games the simulated contestant (who always switches) won 667,049 times—very close to of the time. Thus, it seems that switching doors does make a difference: Switching increases the contestant’s chances of winning. This experiment forces us to reexamine our reasoning. Here is why switching doors is the correct strategy: 2 3 1. When the contestant first made her choice, she had a chance of winning. If she 1 doesn’t switch, no matter what the host does, her probability of winning remains. 3 1 3 2. If the contestant decides to switch, she will switch to the winning door if she had initially chosen a losing one or to a losing door if she had initially chosen the winning one. Since the probability of having initially selected a losing door is, by 2 switching the probability of winning then becomes. 3 2 3 2 We conclude that the contestant should switch, because her probability of winning is 3 if she doesn’t. Put simply, there is a much greater chance that she ini▲ 1 if she switches and 3 tially chose a losing door (since there are more of these), so she should switch. An experiment can be modeled using any computer language or programmable calculator that has a random-number generator. This is a command or function (usually called Rnd or Rand) that returns a randomly chosen number x with 0 x 1. In the next example we |
see how to use this to model a simple experiment. 1 2 3 Contestant: “I choose door number 2.” 1 2 Contestant: “Oh no, what should I do?” E X AM P L E 2 | Monte Carlo Model of a Coin Toss 1 When a balanced coin is tossed, each outcome—“heads” or “tails”—has probability. 2 This doesn’t mean that if we toss a coin several times, we will necessarily get exactly half heads and half tails. We would expect, however, the proportion of heads and of tails to get closer and closer to as the number of tosses increases. To test this hypothesis, we could toss a coin a very large number of times and keep track of the results. But this is a very tedious process, so we will use the Monte Carlo method to model this process. 1 2 To model a coin toss with a calculator or computer, we use the random-number generator to get a random number x such that 0 x 1. Because the number is chosen randomly, 699 699 700 Focus on Modeling PROGRAM:HEADTAIL :0SJ:0SK :For(N,1,100) :randSX :int(2X)SY :J+(1Y)SJ :K+YSK :END :Disp"HEADS=",J :Disp"TAILS=",K A A B 1 2 0 x 1 2B x 1 0 x 1 2 and the outcome “tails” by the event that the probability that it lies in the first half of this interval probability that it lies in the second half “heads” by the event that is the same as the. Thus, we could model the outcome x 1. An easier way to keep track of heads and tails is to note that if 0 x 1, then 0 2x 2, and so 2x, the integer part of 2x, is either 0 or 1, each with probability. (On most programmable calculators, the function Int gives the integer part of a number.) Thus, we could model “heads” with the outcome “0” and “tails” with the outcome “1” when we take the integer part of 2x. The program in the margin models 100 tosses of a coin on the TI-83 calculator. The graph in Figure 1 shows what proportion p of |
the tosses have come up “heads” after n tosses. As you can see, this proportion settles down near 0.5 as the number n of tosses increases—just as we hypothesized. 1 2 1 2 p 1.0 Heads 0.5 0 100 500 Number of tosses FIGURE 1 Relative frequency of “heads” 1000 n ▲ In general, if a process has n equally likely outcomes, then we can model the process using a random-number generator as follows: If our program or calculator produces the random number x, with 0 x 1, then the integer part of nx will be a random choice from the n integers 0, 1, 2,..., n 1. Thus, we can use the outcomes 0, 1, 2,..., n 1 as models for the outcomes of the actual experiment. Problems 1. Winning Strategy In a game show like the one described in Example 1, a prize is con- cealed behind one of ten doors. After the contestant chooses a door, the host opens eight losing doors and then gives the contestant the opportunity to switch to the other unopened door. (a) Play this game with a friend 30 or more times, using the strategy of switching doors each time. Count the number of times you win, and estimate the probability of winning with this strategy. (b) Calculate the probability of winning with the “switching” strategy using reasoning similar to that in Example 1. Compare with your result from part (a). 2. Family Planning A couple intend to have two children. What is the probability that they will have one child of each sex? The French mathematician D’Alembert analyzed this problem (incorrectly) by reasoning that three outcomes are possible: two boys, or two girls, or one child of each sex. He concluded that the probability of having one of each sex is, mistakenly assuming that the three outcomes are “equally likely.” (a) Model this problem with a pair of coins (using “heads” for boys and “tails” for girls), or write a program to model the problem. Perform the experiment 40 or more times, counting the number of boy-girl combinations. Estimate the probability of having one child of each sex. 1 3 (b) Calculate the correct probability of having one child of each sex, and compare this with your result from part (a). 3. Dividing a Jack |
pot A game between two players consists of tossing a coin. Player A gets a point if the coin shows heads, and player B gets a point if it shows tails. The first player to get six points wins an $8000 jackpot. As it happens, the police raid the place when player A has five points and B has three points. After everyone has calmed down, how should the jackpot The Monte Carlo Method 701 be divided between the two players? In other words, what is the probability of A winning (and that of B winning) if the game were to continue? The French mathematicians Pascal and Fermat corresponded about this problem, and both came to the same correct conclusion (though by very different reasonings). Their friend Roberval disagreed with both of them. He argued that player A has probability of winning, because the game can end in the four ways H, TH, TTH, TTT, and in three of these, A wins. Roberval’s reasoning was wrong. 3 4 (a) Continue the game from the point at which it was interrupted, using either a coin or a modeling program. Perform this experiment 80 or more times, and estimate the probability that player A wins. (b) Calculate the probability that player A wins. Compare with your estimate from part (a). 4. Long or Short World Series? In the World Series the top teams in the National League and the American League play a best-of-seven series; that is, they play until one team has won four games. (No tie is allowed, so this results in a maximum of seven games.) Suppose the 1 teams are evenly matched, so that the probability that either team wins a given game is. 2 (a) Use a coin or a modeling program to model a World Series, where “heads” represents a win by Team A and “tails” represents a win by Team B. Perform this experiment at least 80 times, keeping track of how many games are needed to decide each series. Estimate the probability that an evenly matched series will end in four games. Do the same for five, six, and seven games. (b) What is the probability that the series will end in four games? Five games? Six games? Seven games? Compare with your estimates from part (a). (c) Find the expected value for the number of games until the series ends. 7.] 4 P 6 P 5 P four |
games seven be six five Hint: This will 5. Estimating P In this problem we use the Monte Carlo method to estimate the value of p. The circle in the figure has radius 1, so its area is p, and the square has area 4. If we choose a point at random from the square, the probability that it lies inside the circle will be area of circle area of square p 4 The Monte Carlo method involves choosing many points inside the square. Then we have number of hits inside circle number of hits inside square p 4 Thus, 4 times this ratio will give us an approximation for p. To implement this method, we use a random-number generator to obtain the coordinates 2 x, y of a random point in the square, and then check to see if it lies inside the circle (that is, 1 we check if x 2 y 2 1). Note that we need to use only points in the first quadrant, since the ratio of areas is the same in each quadrant. The program in the margin shows a way of doing this on the TI-83 calculator for 1000 randomly selected points. Carry out this Monte Carlo simulation for as many points as you can. How do your results compare with the actual value of p? Do you think this is a reasonable way to get a good approximation for p? PROGRAM:PI :0SP :For(N,1,1000) :randSX:randSY :P+((X 2+Y 2)1)SP :End :Disp "PI IS APPROX",4*P/N y 1 _1 0 1 x _1 702 Focus on Modeling The “contestant’s dilemma” problem discussed on page 699 is an example of how subtle probability can be. This problem was posed in a nationally syndicated column in Parade magazine in 1990. The correct solution was presented in the column, but it generated considerable controversy, with thousands of letters arguing that the solution was wrong. This shows how problems in probability can be quite tricky. Without a lot of experience in probabilistic thinking, it’s easy to make a mistake. Even great mathematicians such as D’Alembert and Roberval (see Problems 2 and 3) made mistakes in probability. Professor David Burton writes in his book The History of Mathematics, “Probability theory abounds in paradoxes that wrench the common sense and trip the unwary.” |
6. Areas of Curved Regions The Monte Carlo method can be used to estimate the area under the graph of a function. The figure below shows the region under the graph of above the x-axis, between x 0 and x 1. If we choose a point in the square at random, the probability that it lies under the graph of is the area under the graph divided by the 2 area of the square. So if we randomly select a large number of points in the square, we have, number of hits under graph number of hits in square area under graph area of square Modify the program from Problem 5 to carry out this Monte Carlo simulation and approximate the required area. y 1 0 y=≈ 1 x 7. Random Numbers Choose two numbers at random from the interval probability that the sum of the two numbers is less than 1? (a) Use a Monte Carlo model to estimate the probability. 0, 1 3 2. What is the (b) Calculate the exact value of the probability. [Hint: Call the numbers x and y. Choos- ing these numbers is the same as choosing an ordered pair x, y. What proportion of the points in this square corresponds in the unit square 2 1 x, y 0 x 1, 0 y 1 5 1 to x y being less than 1?] 2 0 6 ANSWERS TO SELECTED EXERCISES AND CHAPTER TESTS CHAPTER P SECTION P.1 ■ page 5 1. 48 3. T $9.60 9. (a) 38 km3 11. (a) (b) 2 km3 Depth (ft) Pressure (lb/in2) 5. $300 7. (a) 30 mi/gal (b) 7 gal 0 10 20 30 40 50 60 14.7 19.2 23.7 28.2 32.7 37.2 41.7 A a b 2 (b) C 12 n (b) 34 ft 13. N 7„ 15. 17. C 3.50x 21. (a) $15 19. d 60t 23. (a) (i) C 0.04x 25. (a) $2 (ii) $1200 (d) C F rt (c) 12 min (c) 4 (ii) C 0.12x (b) C 1.00 0.10t (b) (i) $400 SECTION P.2 ■ page 12 (b) 3 (c) |
12 22 7 3. Denominator 3 (d) 2 (c) 0, 10, 50, 1. (a) 2 (b) 0, 10, 50 7. 1 9. 2 13. Associative Property for Addition 17. Commutative Property for Multiplication 19. 3 x 21. 4A 4B 23. 3x 3y 11. Commutative Property for Addition 25. 8m 27. 5x 10y, 0.538, 1.23, (d) 1 3 15. Distributive Property 29. 17 30 31. 1 15 33. 3 35. 13 20 37. 8 3 39. 15 2 41. (a) 7 9 (b) 13 45 (c) 19 33 43. Distributive Property SECTION P.3 ■ page 17 1. Each positive number x is represented by the point on the line a distance of x units to the right of the origin, and each negative number x is represented by the point x units to the left of the origin. 2 x 7 3. 2, 7 2 (b) (c) 9. False 17. 5. absolute value; positive 11. True 19. 7. (a) 15. False 13. False ; x 6 5 1 0 −3 1 2 21. x 1 23. x 3 2 25. (a) x 0 (b) t 4 (c) a p (d) 5 x (b) 2, 4, 6 5 6 x 5 x 31. (a) 5 33. 3 x 0 0 1 (e) 3 29. (a) 5 p 3 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 27. (a) 5 0 0 5 (b) x 5 0 1 x 4 6 35. 2 x 8 1, 2, 3, 4, 5, 6, 7, 8 6 (b) 7 6 5 6 6 0 2 1 39. q, 1 4 8 1 −3 37. x 2 2 1 41. 43. 1 45. (a) 2, 1 4 1, q 2 3, 5 47. 49. 51. 3 −2 0 −4 1 −2 −1 (b) 4 3, 5 1 4 1 6 4 (b) 73 (b) 24 (b) 1 67 40 57. (a) 12 (b) 5 55. (a) 2 (c) 61. (a) 15 TG: 9, |
3, 0, 5, 8, 1, 1 TG 0 TG gives more information because it tells us which city 53. (a) 100 59. 5 63. TO TO 0 TO had the higher (or lower) temperature. (b) 6 ft : 9, 3, 0, 5, 8, 1, 1 65. (a) Yes, No 5. (a) 50 17, 13 2 SECTION P.4 ■ page 24 1. 56 13. 9 3. Add, 39 5. Multiply, 38 7. 125 21. 16 17. 1000 19. 15. 1 2 9. 64 1 23. 4 11. 1 25. 1 27. 3 8 29. x10 31. 33. 1 35. y3 37. a6 39. 8y6 41. a18 1 9 1 x4 43. 1 24z4 45. a6 64 47. 8x7y5 49. 6a3b2 51. 405x10y23 65. 61. 59. 69. 67. 63. b3 3a √10 u11 3y2 z 4a8 b9 53. s2t7 a19b c9 55. 8rs4 57. y2z9 2 x s3 r4q7 75. 2.8536 105 77. 1.2954 108 81. 319,000 87. 0.00855 (c) 3.3 1019 molecules 91. 1.3 1020 93. 1.429 1019 95. 7.4 1014 97. (a) Negative 79. 1.4 109 85. 710,000,000,000,000 (b) 4 1013 cm 83. 0.00000002670 89. (a) 5.9 1012 mi 73. 6.93 107 (b) Positive 125 x6y3 71. (c) Negative A1 A2 Answers to Selected Exercises and Chapter Tests (e) Positive (d) Negative 101. 1.3 1021 L 103. 4.03 1027 molecules 105. $470.26, $636.64, $808.08 (f) Negative 99. 2.5 1013 mi SECTION P.5 ■ page 30 1. 51/3 3. No 5. 9. 23 42 17. (a) 2 3 11. 53/5 (b) 4 1 13 1 13 # 13 13 13 3 7. 51/2 13. 25 a2 |
15. (a) 4 3 2 (b) 4 (b) 2 (c) 1 5 (c) 19. (a) 1 2 21. (a) 1 4 (c) 1 2 (b) 36 (c) 100 25. 5 27. 14 (b) 29. 41. 1 81 (c) 4 23. (a) x 0 0 215 31. 2x 2 33. 43. 13 4 45. 1 1000 x13 y 215 35. 47. 53. „ 5/3 55. 4a4b 57. 1 4y2 59. ab25 ab2 315 3 x3 y1/5 37. 2 49. x2 0 61. 4st4 7 12 39. x 0 51. 16b3/4 2y4/3 x2 63. 65. 77. 1 x 67. x1/4y1/4 2 x ab10y10/3 79. 4u √2 69. y3/2 71. 10x7/12 73. 2st11/6 75. x 81. y1/2 83. (a) 26 6 (b) 322 2 (c) 323 85. (a) (b) 25 x3 x (c) 27 x4 x 23 2 2 (b) 24 27 3 (c) 425 16 87. (a) 23 x2 x 89. 41.3 mi 91. (a) Yes (b) 3292 ft2 13. No 15. Yes; 3 21. 5x2 2x 4 5. A2 2AB B2, 41. 21t2 26t 8 3. like, x3 8x2 5x 2 23. x3 3x2 6x 11 27. 2x2 2x 29. x3 3x2 31. t2 4 35. 2x4 x3 x2 37. x2 x 12 43. 6x2 7x 5 SECTION P.6 ■ page 36 1. (a), (c) 4x2 12x 9 7. Trinomial; x2, 3x, 7; 2 9. Monomial; 8; 0 11. Four terms; x4, x3, x2, x; 4 17. No 19. 7x 5 25. 9x 103 33. 7r3 3r2 9r 39. r2 2r 15 45. 2x2 5xy 3y2 51. 9x2 24x |
16 57. x4 2x2 1 65. x 4 71. x3 4x2 7x 6 77. y y2 79. x4 2x2y2 y4 85. 1 x4/3 91. (b) 4x 3 32x 2 60x; 3 93. (a) 2000r 3 6000r 2 6000r 2000; 3 (b) $2122.42, $2185.45, $2282.33, $2382.03, $2662.00 49. x2 6x 9 55. 4x2 12xy 9y2 63. x2 9y2 69. 1 6r 12r2 8r3 75. x3/2 x 59. x2 25 67. y3 6y2 12y 8 47. 6r2 19rs 10s2 53. 4u2 4u√ √2 73. 2x3 7x2 7x 5 89. 4x2 4xy y2 9 87. x4 x2 2x 1 61. 9x2 16 81. x4 a4 83. a b2 (c) 32, 24 49. 55. 61. 65. 69. 73. 77. 81. 85. 89. 1 1 x 6 2 1 2x y 1 t 1 2 1 x2 2y 1 x 3 x2 x3/2 x1/3 1 2x 1 51. 7 2y x 6 2 2 57. 4ab 2 t2 t 1 1 59. 63. 2 1 x4 2x2y 4y2 x 1 2 x2y3 2 1 t 3 2 2 x 1 7 2y 2 1 x 3 2 1 1 2x 5 2 1 x 67. x y 2 1 53. 1 x 1 2 x 3 2 1 4x2 10x 25 /3 2 71. 2 y 3 79. a 1 2 1 87. 1 3x 4 1/2 y4 1 a 1 2 1 75. 1 y 2 2 1 2x2 1 3 2 1 y 1 2 1 x1/2 83. 2 1/2 x2 1 2 2 1 16x2 91. 2 1 x2 3 93. 1 97. (d 7x 35 x2 3 5x 9 4/3 2 3x2 95 SECTION P.8 ■ page 52 1. (a), (c) 3. (a) False (b) True 5. (a) |
3 (b) x 7. (a) 53 (b) 9. (a) 5 (b) x (c) 2x2 1 x 1 2 x 1 11. (a) 3 2 (b, x 2 6 23. x 2 x 1 (b) x 5 0 1 x 2 31. x 3 x 3 2 x yz 45 13. (a 29. 0 5 5y 10 y 19. 27. 35. x 2x 3 1 2x 3 x 4 x 1 37. 21 2x 3 1 2 1 x 4 2 41. x2 x 1 1 2 43. 2 x 17. y y 1 1 t2 9 2x 1 1 2x 1 2 2 1 x 5 1 3x 7 2 55. 49. 1 x 1 2x 1 x 1 x2 2 5x 6 x 1 x 1 1 61. 1 x 2 2 2 1 3x 2 x 1 2 2 51. 1 2x 7 57 u2 3u 67. 71. xy 73. 2 2 1 x 1 x2 2x 1 x2 y2 x y xy 75 4x 63. 2 1 69. x 1 1 1 1 x 2x h x h 2 x2 2 77. 83 13 85. 79. 1 x 2 x 1 3/2 2 89. 2 23 91. 2 93. 1 27 22 5 2 1 97. 81. 1 21 x2 87. 2x 3 x 1 4/3 2 1 y23 y2y 3 y r 2 2r 22 B 5 A 99. 1 2x 2 1 x 105. False 107. True (b) 20 3 6.7 ohms 15. 25. 33. 39. 47. 53. 59. 65. 2 1 5 2 1 17. 1 11. 2x3; 2x3 5. SECTION P.7 ■ page 43 1. 3; 2x5, 6x4, 4x3; 2x 5 1 9. xy 2x 5 2 2x 6y 3x 8 3x 4 2 1 9x2 3xy y2 3x y 2 1 x 6 2 29. x2 15. 19. 23. 27. 33. 39. 45. x 4 35. 1 21. 47. 41 x2 3x 2 2 7. 2x 3x 1 3. 1 x2 8 13 3a 4 25. 2 x2 1 2 1 6x 1 y 3 2 1 3x 2 |
2 2x2 3 1 2 1 1 2 3a 4 2 1 2s 5t 31. 1 2 2 1 2x 1 3y3 1 37. 2 y 5 2 2x 3 43. 2 4s2 10st 25t2 x2 3 2 95. 2 2 1 2y 5 1 2x 25 3 B A 101. True 109. (a) 103. False R1R2 R2 R1 x A 2 1x B 4 x x 3y 2xy 7 5/2 4r s x 6 59. 67. 61. 7.825 1010 63. 1.65 1032 x 3 69. 3x 1 x 1 2 2 1 1 77. 5.06 79. 43.66 81. 1.60 CHAPTER P REVIEW ■ page 56 1. (a) T 250 2x number, integer number, integer (f) rational, integer 7. Distributive Property 13. 2 x 6 (b) 190 (b) rational, integer (d) irrational −2 6 (c) 125 3. (a) rational, natural (c) rational, natural (e) rational, neither 5. Commutative Property for Addition 1 (b) 6 11. (a) 9. (a) (b) 3 2 9 2 15. x 4 5, q 2 1, 5 17. 19. 3 1 21. (a) 4 5 −1 4 1, 0, 1 5 2, 1, 2, 3, 4 27. 6 6 5 (b) 23. (a) 1, 2 5 33. 11 6 35. 5 1 5 6 31. 5 6 (b) 25. 3 3 5 1 2, 1 5 37. (a) (b) 41. x 4m2 43. x abc 45. x 5c1 47. (a) 71/3 49. (a) x5/6 1 29. 72 3 1 6 8 (b) x9/2 2 2 0 1 0 0 0 51. 12x 5y 4 53. 9x 3 55. x 2y 2 39. x 2 (b) 74/5 57. 65. 71. 75. 79. 1 4t 73. 2 2 x 1 2 3 4x 4t 2 5 4t 2x x 2 2 1 81. 77 83. 89. 4a 4 4a 2b b 2 91. x 3 6x 2 11x 6 85 87. |
6x 2 21x 3 2 93. 2x 3/2 x x 1/2 95. 2x 3 6x 2 4x 97. x 3 2x 3 99. 3 x 3 1 x 4 2 101. x 1 x 4 107. x 113. 5 0 119. No 109. 6x 3h 5 1 2x x 0 and x 4 121. No 103. 1 x 1 111 10 6 115. No 117. Yes 6 105. 1 x 1 CHAPTER P TEST ■ page 58 1. (a) C 9 1.50x (b) Irrational integer 3. (a) 0, 1, 5 integer (b) $15 (c) Rational, integer 2, 0, 1 (b) 5 6 5 2, 1, 3, 5, 7 4. (a) −4 2 0 2. (a) Rational, natural number, (d) Rational, (b) Intersection [0, 2) Union [4, 3] 5. (a) 64 1 (h) 27 −4 (c) (b) 64 6. (a) 1.86 1011 (b) 48a 5b 7 (c) 5x 3 0 2 3 6 (c) 0 0 4 2 9 1 (g) 2 16 622 4 9 1 64 1 49 (f) (d) (e) (b) 3.965 107 7. (a) x 9y (c) a b (d) 4x 2 12x 9 8. (a) 11x 2 (d) 7 (b) 4x 2 7x 15 (e) x 3 6x 2 12x 8 (f) x4 9x2 9. (a) (b) x 3 x 2 1 x 2 2x c) 2 2 3x 9 x 1 (e) 2 (d) 2 1 3x1/ 2x 5 2x 5 2 2 1 Answers to Section 1.3 A3 (f) (d) x 2 x y xy 1 1 x 2 2 11. (a) 2 1 2 10. (a) 313 2 (b) x 1 x 3 (c) 1 x 2 x 2 x 2 (b) 216 3 12 25 32 FOCUS ON PROBLEM SOLVING ■ page 62 1. 37.5 mi/h 3. 150 mi 11. 2p 15. 15,999,999,999,992,000 |
,000,000,001 7. 75 s 5. 427 9. The same amount CHAPTER 1 SECTION 1.1 ■ page 72 3. (a) True 1. (a), (c) (c) False 5. (a) No (b) Yes 11. (a) Yes 19. 3 23. 21. 12 35. 33. 3 1 2 47. No solution 2 12 65. No solution 55. (b) False (because quantity could be 0) (b) Yes 7. (a) Yes (b) No 3 25. 4 13. 12 32 9 15. 18 29. 4 9 37. 39. 13 3 49. No solution 27. 30 29 41. 2 51. 7 57. No solution 67. 5, 1 59. 4, 0 69. 8 71. 125 R PV nT 83. (b) No 9. (a) No 31. 20 17. 9 1 3 45. 2 2 16 63. 2 43. 3 97 53. 61. 3 73. 8 85. „ 1 75. 3.13 P 2l 2 2 1/2 x 1 x 1 2 2 87. x 2d b a 2c 89. x 1 a a2 a 1 91. r B 3V ph 93. b 2c2 a2 95. r 3 3V B 4p 97. (a) 0.00055, 12.018 m (b) 234.375 kg/m3 99. (a) 8.6 km/h (b) 14.7 km/h SECTION 1.2 ■ page 82 3. (a) x2 (b) l„ (c) pr2 5. 1 x 7. 3n 3 9. 160 s 3 11. 0.025x 13. A 3„2 15. d 3 4 s 17. 25 x 3 27. $45,000 29. Plumber, 70 h; assistant, 35 h 1 19. 400 miles 21. $9000 at % and $3000 at 4% 23. 7.5% 4 2 25. $7400 31. 40 years old 33. 9 pennies, 9 nickels, 9 dimes 37. 120 ft by 120 ft 45. 200 mL 47. 18 g 55. 3 h 63. 120 ft 59. 500 mi/h 61. 6.4 ft from the fulcrum 49. 0.6 L 51. 35% 53. 37 min 20 s |
39. 8.94 in. 65. 18 ft 35. 45 ft 41. 4 in. 43. 5 m 57. 4 h 6 3 SECTION 1.3 ■ page 94 1. (a) b 2b2 4ac 2a (b) 1 2, 1, 4; 4, 2 3. b2 4ac; two distinct real; exactly one real; no real 11. 3, 4 13., 2 9. 3, 1 2 1 3 1 2 15. 20, 25 17. 5. 4, 3 7. 3, 4 19. 3 2 15 21. 1 2, 3 2 23. 21, 1 25. 2 1 16 114 2 27. 0, 1 4 29. 3, 5 31. 2, 5 33. 3 2, 1 35. 3 15 2 37. 6 3 17 39. 45. No real solution 3 2 16 3 15 1 2 47. 41. 3 4 43. 9 2, 1 2 49. 8 114 10 A4 Answers to Selected Exercises and Chapter Tests 51. No real solution 57. 0.985, 2.828 53. 0.248, 0.259 2√ √0 2 0 g t 59. 55. No real solution 2gh 11. q, 7 2 4 1 61. x 2h 24h2 2A 2 63. s a b 2c 1 2 2a2 b2 4c2 2ab 2 65. 2 67. 1 69. No real solution 71. 2 73. 1 a 75. 1, 1 1 a 83. 60 ft by 79. 19 and 36 81. 25 ft by 35 ft 77. k 20 85. 48 cm 87. 13 in. by 13 in. 40 ft 91. 50 mi/h (or 240 mi/h) 93. 6 km/h 97. (a) After 1 s and (b) Never 1 (e) After 99. (a) After 17 yr, on Jan. 1, 2019 2 2 (b) After 18.612 yr, on Aug. 12, 2020 103. Irene 3 h, Henry 180 ft 1 1 2 h s s 105. 215,000 mi 89. 120 ft by 126 ft 95. 4.24 s (c) 25 ft (d) After 101. 30 ft; 120 ft by 1 4 2 1 1 4 s SECTION 1.4 ■ page 102 3. (a) 3 4i 1. 1 imaginary part 7 7. Real part |
(b) 9 16 25 2 3 5. Real part 5,, imaginary part 5 3 13 2 3 9. Real part 3, imaginary part 0 11. Real part 0, imaginary 13. Real part part 17. 3 5i 19. 2 2i 25. 30 10i 35. i 37. 45. i 27. 33 56i 29. 27 8i 39. 5 12i 1 5 i 49. 6, imaginary part 2 15. 5 i 21. 19 4i 23. 4 8i 31. i 41. 4 2i 3 15 3 15 47. 5i 43. 51. 8 5 i 33. 1 2 4 3 i 53. 2 1 2 1 2 55. i 12 57. 7i 59. 2 i 61. 1 2i 63. 1 2 13 2 i 65. 1 2 1 2 i 67. 3 2 13 2 i 69. 6 16 i 6 71. 1 3i SECTION 1.5 ■ page 110 1. (a) 0, 4 5. 0, 4 12 7. 0, 3 (b) factor 17. 2 15., 5 11. 0, 2, 3 3. quadratic; x 1; W2 5W 6 0 2 12 23. 4 35. 7, 0 21. 50, 100 33. 6 31. 4 13. 0, 9. 0, 2 19. 1.4, 2 27. 2 29. 4 2 12, 15 41. No real solution 43. 1, 3 25. No real solution 2, 37. 3 3 4 3 13, 2 12 39. 45. 47. 1, 0, 3 49. No solution 51. 27, 729 63. 53. 1 2 55. 20 57. 3, 61. 1, 1 i 13 2 63. 0, 1 113 2 1 i 13 2 59. 2 65. 12, 2 67. 1, 2, 1 i 13 2 2a2 36, 1 i 13 69. 3i 71. i 1a, 2i 1a 75. 50 73. 83. 16 mi; No 85. 49 ft, 168 ft, and 175 ft 77. 89 days 79. 7.52 ft 87. 132.6 ft 81. 4.63 mm SECTION 1.6 ■ page 119 1. (a) (b) (c) (d) 3. 5. 4, 2, 4 12 7. 9. 5 2, 1, 2, 2 4 1, q 2 1 q |
, 18 2 −18 3, 1 2 −3 9 2, 5 2 9 2 2, 3 −2 3, 6 −3 1, 4 −1 2, 2 −2 2 4 2 2 7 2 2 −1 5 3 6 4 2 −2 4 2, 0 1 −2 2, q 1 2 2 0 2 59. q, 3 2 2 1 − 3 2 2, q 1 2 2 1, 3 1 4 2, 0 −2 2, 0 0 2 2 − 15. 19. 23. 27. 31. 35. 39. 43. 47. 51. 55. 67. 71. 73. 75. (b) 13. 17. 21. 25. 4, q 2 q, 1 2B 16 3, q B 1 4 A A 16 3 q, 1 1 −1 4 29. 2, 6 2 1 2 33. 5 2, 11 11 2 37. 1 q, 7 2 4 41. 45. 49. − 7 2 q, 1 1 −1 q, 3 1 −3 q, 2 1 −2 1 −1 q, 1 0, q 1 2, q 6, q 2 2 2 1, 3 4 3 57. 61. 65. 1 −1 1 5 3 −2 69. A −3 q, 5 3 2 16, q 2 16 2, 1 −1 0, 1 1 4 1 2 0 3, 1 2B 1 2, q 2 − 1 2 −2 2 1 1 2 q, 2 q, 1 1, 2 2 1, q 3 1 1 2 4 2 2, q 2 −1 x 4 3 77. x 2 or x 7 79. (a) x 2a c 81. 68 F 86 4 3 a c b b 83. More than 200 mi 85. Between 12,000 mi and 14,000 mi x c a c b 87. (a) 1 3 P 560 3 (b) From $215 to $290 q, 2 1 2 2, 4 2 53. 1, 3 4 3 89. Distances between 20,000 km and 100,000 km 91. From 0 s to 3 s 93. Between 0 and 60 mi/h 95. Between 20 and 40 ft 5 2, 3 7. (a) 1 8. 41 F to 50 F 9. 0 x 4 0, 1 (b) 1 1 2 4 2, q (c) 1, 5 1 2 2 (d) 3 4, 1 2 Answers |
to Section 2.1 A5 3. SECTION 1.7 ■ page 124 1. 3, 3 3, q 1 11. 4.5, 3.5 1 2 2, 35 25 q, 3, 3 13. 4, 2, 1 4 q, 2 2, 8 19. 21. 29. 2 3 4 27. 31. 37. 3 1 1 4 q, 7 1 4 6.001, 5.999 3 3, q 4 2 39. B 2 3, q A 7, 13 2 B 3 1 A 47. 43. 15 q, 1 2B 2, 7 2 53. x 0 (b) 0.017 x 0.023 A 55. x B A 0 0 0 9. 1, 5 5. 6 2 15. 3, 1 23. [4, 4] 25. 7. 5 17. 8, 2 q, 7 2B A 7 2, q A B 1.3, 1.7 35. 4 4, 8 1 2 0, q 2 3 33. 3 6, 2 2 4, 1 41. 1 2, 3 2 4 1, 4 3 1 45. 3 49. x 0 0 4 3 3 51. 57. (a) 0 x 0.020 4 0 0 x 7 5 0 0.003 CHAPTER 1 REVIEW ■ page 126 1. 5 3. 4 5. 5 7. 15 2 9. 6 11. 0 13. 12 15. 2 14 5 17. 13 3 19. x 2A y 21. t 11 6J 23. 5 25. 27 27. 625 29. No solution 31. 2, 7 33. 1, 1 2 35. 0, 5 2 37. 2 17 3 47. 2, 7 3 16 3 49. 20 lb raisins, 30 lb nuts 41. 3 39. 43. 1 45. 3, 11 51. 1329 3 1 41 3.78 mi/h 53. 12 cm, 16 cm 55. 23 ft by 46 ft by 8 ft 57. 3 9i 59. 19 40i 61. 65. 1 2 213 2 71. 4, 4i 73. i 2 1 2 213 13 3 i 5 12i 13 67. 4i 63. i 69. 3 i 3, q 2 −3 q, 6 −6 q, 2 1 1 2 2 2, q 2 2 2, 4 4 −2 2 4 75. 79. 83. 87. 1 1 1 1 q, 1 |
0, q 3 2 4 3, 8 3 4 89. (a) 3 (b) 0, 1 1 2 77. 81. 85. 3, 1 1 −3 4, 1 3 −4 4 2 2, 8 4 3 2 −1 −1 −1 8 0 FOCUS ON MODELING ■ page 133 1. (a) C 5800 265n (c) (b) C 575n n Purchase Rent 12 24 36 48 60 72 8,980 12,160 15,340 18,520 21,700 24,880 6,900 13,800 20,700 27,600 34,500 41,400 (d) 19 months (c) P 27x 8000 7. (a) Minutes used 3. (a) C 8000 22x (b) R 49x 5. (a) Design 2 (b) Design 1 (d) 297 Plan A Plan B Plan C $30 $80 $130 $180 $230 $280 $330 $40 $70 $100 $130 $160 $190 $220 $60 $70 $80 $90 $100 $110 $120 500 600 700 800 900 1000 1100 2, x 500 B 40 0.30 A 30 0.50 1 x 500 (b) 2 1 (c) 550 minutes: A $55, B $55, C 60 0.10 C $65; 975 minutes: A $267.50, B $182.50, C $107.50; 1200 minutes: A $380, B $250, C $130 (d) (i) 550 minutes (ii) 575 minutes x 500, 1 2 2 CHAPTER 2 SECTION 2.1 ■ page 142 2 c a 3, 5 2 1. 3. 1 2 1 2 d b 1 2; 10 2 y 5 (−4, 5) (4, 5) (−2, 3) (2, 3) −5 0 5 x −5 (−4, −5) (4, −5) 5. 7. CHAPTER 1 TEST ■ page 129 1. (a) 5 (b) 5 2 (c) 512 (d) 15 2 2. c B 4. (a) 5 4i (b) 1 6i (c) 7 3i 3. 150 km E m 2 5i (e) i _5 (f) 212 8i 5. (a) 3, 4 (b) (c) No solution 1 5 (d) 2 |
i 12 2 (d) 1, 16 (e) 0, 4 (f) 2 3, 22 3 6. 50 ft by 120 ft 9. 5 x _5 y 5 0 _5 y 5 0 _5 5 x A6 Answers to Selected Exercises and Chapter Tests 11. 13. 33. (a) 35. 24 y 5 0 _5 _5 x5 _5 15. 17. y 5 0 _5 y 2 y 5 0 _5 y 4 0 −4 19. _5 −4 5 x −2 0 −2 2 x 4 x 21. (a) 113 (b) 25. (a) 3 2, 1 B A 23. (a) 10 (b) 27. (a) 1, 0 1 2 y (6, 16) 8 (0, 8) (4, 18) y 6 −8 0 8 x −6 0 6 x (b) 10 29. (a) (c) 3, 12 1 2 y 4 (_1, 3) _4 0 4 x _4 (6, _2) (−3, −6) (b) 25 31. (a) (c) 1 2, 6 A B y 4 0 (11, 6) (7, 3) 4 x (b) 174 (c) 5 2, 1 2B A (b) 5 (c) 9, 9 2B A y 5 0 (3, 4) 5 x 5 x _5 (_3, _4) _5 y 5 A(1, 3) B(5, 3) _3 0 3 x C(1, _3) D(5, _3) _5 (b) 10 (c) 37. Trapezoid, area 9 0, 0 1 2 y 5 D C _3 0 A 3 B x _5 39. 53. A 6, 7 2 1 2, 3 1 2 41. Q 1, 3 1 2 45. (b) 10 49. 0, 4 1 2 51. 1, 7 2B A y 2 0 Q(1, 1) R(4, 2) 5 x (2, _3) _5 P(_1, _4) 55. (a) y 4 D C B 0 4 x _4 A _4 (b) 5 2, 3 5 2, 3, A B B A 57. (a) 5 (b) 31; 25 same street or the same avenue. 59 |
. midpoint is the pressure experienced by the diver at a depth of 66 ft. (c) Points P and Q must be on either the ; the y-value of the 66, 45 2 1 SECTION 2.2 ■ page 154 5. No, no, yes 7. No, yes, yes 1, 2 ; 3 2 1 1. 2; 3; No 3. 9. Yes, no, yes 11. x-intercept 0, y-intercept 0 y 2 −2 0 2 x −2 13. x-intercept 4, y-intercept 4 y 2 0 2 x Answers to Section 2.2 A7 15. x-intercept 3, y-intercept 6 17. x-intercepts 1, y-intercept 1 35. x-intercept 0, y-intercept 0 y 2 0 −4 −4 4 x −5 y 1 0 − 19. x-intercept 0, y-intercept 0 21. x-intercepts 3, y-intercept 9 y 5 y 2 − 6 0 −2 6 x −5 0 x 5 23. No intercepts 25. x-intercept 0, y-intercept 0 y 4 0 − 4 y 4 53. 3 27. x-intercepts 2, y-intercept 2 29. x-intercept 0, y-intercept 0 y 3 0 −5 5 x −5 y 5 0 −5 31. x-intercepts 4, y-intercept 4 33. x-intercept 0, y-intercept 0 5 x 55. 59. 63. 69. 73. 1 1 1 A 37. x-intercepts 0, 4; y-intercept 0 39. x-intercepts 2, 2; y-intercepts 4, 4 41. x-intercept 3; y-intercept 3 43. x-intercepts 3; y-intercept 9 45. x-intercepts 2; y-intercepts 2 49. 0, 0 47. None 3, 0 51 71. 2 4, 2 9 2 25 2 4 1, 1 4B 2 A y 10 (_2, 5) x2 y2 65 2 x 7 1 1, 2 2, 2 1 67. 57. 61. 65. 1 2 y 3 2 9 2 2, 5 1, 4 2 75. (_3, 6) y 3 y 5 0 −2 −2 |
_10 0 4 x _3 0 _1 3 x 77. Symmetry about y-axis 81. Symmetry about x-axis, y-axis, and origin 79. Symmetry about origin 1.2 2 9 2 A8 Answers to Selected Exercises and Chapter Tests 83. 85. y 1 0 −1 −4 4 x −4 87. 892 −2 2 x − 4 −2 0 2 4 x −2 − 4 27. −6 4 −4 29. 6 −1.2 0.8 −0.8 5 14 31. 4 33. 2.5, 2.5 39. 43. 3.00, 4.00 49. 1.00, 0.00, 1.00 57. 2.05, 0, 1.05 q, 1.00 61. 1 q, 0 65. 1 73. (a) 20 2 1 12 35. 4 5.7 37. No solution 5 2 14 5 7.99, 5 2 14 5 2.01 41. 45. 1.00, 2.00, 3.00 47. 1.62 51. 0, 2, 3 53. 4 55. 2.55 59. 2.00, 5.00 3 [2.00, 3.00] 63. 4 1, 4 67. 69. 2 4 1.00, 0 1 1, 3 3 (b) 67 mi 4 1.00, q 71. 0, 0.01 2 1 91. 12p 93. (a) 14%, 6%, 2% (b) 1975–1976, 1978–1982 (d) 14%, 1% (c) Decrease, increase 0 100 3. (a) x 1, 0, 1, 3 SECTION 2.3 ■ page 164 1. x 7. (c) 9. (c) 11. (b) 1, 0 4 1, 3 3 4 3 5. (c) 13. 400 20 SECTION 2.4 ■ page 176 x 1 5. 15. 11. 9. 17. 3x 2y 6 0 1 2 1. y; x; 2 3. 13. 2, 1, 3, 2 19. 5x y 7 0 23. 5x y 11 0 27. 3x y 3 0 33. x 1 39. (a) 21. 2x 3y 19 0 25. 3x y 2 0 29. y 5 35. 5x 2y 1 0 |
31. x 2y 11 0 37. x y 6 0 (b) 3x 2y 8 0 −10 2 −2 15. −20 19. −4 5 −1 100 −50 23. No 25. Yes, 2 −4 17. −50 21. 20 6 −3 10 150 5 −10 2000 −2000 5 −1 y 5 (−2, 1) 0 1 x −3 41. They all have the same slope. 43. They all have the same x-intercept. 8 5 m = 1.5 m = 0.75 m = 0.25 m = 0 8 m = −0.25 m = −0.75 −5 m = −1.5 − 47. 1 3, 0 y 2 0 −2 −5 5 x −5 45. 1, 3 y 5 0 −2 b = −3 −8 b = −6 b = −1 5 x 49. 3 2, 3 51. 0, 4 y 1 0 −2 2 x −5 53. 3 4, 3 55. 3 4, 1 4 y 1 0 −3 −2 0 −1 2 x 63. (b) 4x 3y 24 0 61. x y 3 0 65. 16,667 ft dosage for a one-year increase in age. 69. (a) y 67. (a) 8.34; the slope represents the increase in (b) 8.34 mg (b) The slope represents production cost per toaster; the y-intercept represents monthly fixed cost. 12000 9000 6000 3000 0 500 1000 1500 x (b) 76F t 5 24 n 45 71. (a) 73. (a) P 0.434d 15, where P is pressure in lb/in2 and d is depth in feet (b) y 5 0 5 x (c) The slope is the rate of increase in water pressure, and the y-intercept is the air pressure at the surface. 75. (a) (d) 196 ft C 1 4d 260 y (b) $635 (c) The slope represents cost per mile. (d) The y-intercept represents monthly fixed cost. 1000 500 0 500 1000 x SECTION 2.5 ■ page 182 1. directly proportional; proportionality 3. directly proportional; inversely proportional 9. y ks/t 7. √ k/z 5. T kx 11. z k1y |
13. V kl„h 15. R k i Pt 17. y 7x Answers to Chapter 2 Review A9 21. M 15x/y s 500/ 1t 19. R 12/s 27. 31. (a) C kpm (b) 0.125 (c) 324 (b) 0.012 41. (a) R kL/d 2 43. (a) 160,000 (b) Halves it 23. W 360/r 2 25. C 16l„h 29. (a) F kx (b) 8 (c) 32 N (c) $57,500 33. (a) P ks 3 35. 0.7 dB 37. 4 (b) 0.002916 39. 5.3 mi/h (c) R 137 45. 36 lb 47. (a) (b) 1,930,670,340 f k L CHAPTER 2 REVIEW ■ page 187 1. (a) y Q(3, 7) P(0, 3) 1 0 1 x (b) 5 (d) (c) m 4 3 3 2, 5 A B ; point-slope: slope-intercept: y 4 3x 3 ; y 7 4 31 (e) x 3 ; 2 x2 (4, −14) 3. (a) P(−6, 2) (d) m 8 5 y 3 2 1 2 25 y 2 0 (3, 7) (0, 3) 2 x (b) 2 189 (c) 1, 6 1 2 slope-intercept: ; point-slope: y 8 5x 38 5 y 14 8 51 x 6 (e 356 y 4 0 y 8 4 x (−6, 2) −8 0 8 x −8 (4, −14) A10 Answers to Selected Exercises and Chapter Tests 5. 7. B 9. y 3 0 x 5 1 11. (a) Circle 31. 10 x 5 _2 8 _10 33. _3 10 _25 6 2 2 1 y 1 2 2 26 (b) Center, radius 1 1, 3 1 2 y 1 0 1 x 35. 1, 7 37. 2.72, 1.15, 1.00, 2.87 41. (b) (b) 1.85, 0.60 2 1 2x y 6 0 2x 3y 16 0 39. 3 1, |
3 y 2x 6 4 3 (b) x 3 0 0.45, 2.00 1 45. (a) 2 y 2 43. (a) 3x 16 47. (a) x 3 2x 5y 3 0 y 2 5x 3 5 (b) 49. (a) 51. (a) y 2x 53. (a) The slope represents the amount the spring lengthens for a one-pound increase in weight. The S-intercept represents the unstretched length of the spring. 2x y 0 (b) 17. y 7 0 _7 21. 7 y 2 x 5 x 4 0 _4 _3 3 x (b) 4 in. 55. M 8z 57. (a) I k/d 2 (b) 64,000 59. 11.0 mi/h 61. F 0.000125q1q2 (c) 160 candles 63. x 2 y 2 169, 5x 12y 169 0 x CHAPTER 2 TEST ■ page 189 1. (a) y Q(7, 5) x 2 0 2 P(1, _3) (b) 10 4, 1 (c) x 4 1 2 2 y 1 (d) 4 (e) 3 2 25 2 1 2 (f) 1 2. (a) ; 5 2 2 1 0, 0 y y 3 4x 4 (b) ; 3 2 3 13. (a) No graph 15. y 2 0 _2 y 2 0 _5 19. 23. _2 _5 y 2 0 2 x 3, 1 (c 25. (a) Symmetry about y-axis y-intercept 9 27. (a) Symmetry about y-axis y-intercepts 0, 2 29. (a) Symmetry about x- and y-axes and the origin (b) x-intercepts 4, 4; no y-intercept (b) x-intercepts 3, 3; (b) x-intercept 0; 3. (a) symmetry about x-axis; x-intercept 4; y-intercepts 2, 2 y 1 0 1 x (b) No symmetry; x-intercept 2; y-intercept 2 y 1 0 1 x 4. (a) x-intercept 5; y-intercept 3 (b) y 2 0 2 x (d) y 3 5x 3 3 |
(c) 5 5. (a) 3x y 3 0 6. (a) 4C 5 3 (e) (b) 2x 3y 12 0 (b) T 1 0 10 x 100 (c) The slope is the rate of change in temperature, the x-intercept is the depth at which the temperature is 0C, and the T-intercept is the temperature at ground level. 7. (a) 2.94, 0.11, 3.05 (b) (c) 12,000 lb 8. (a) M k „h2/L (b) 400 1.07, 3.74 4 3 y 0 −4 Answers to Chapter 2 Focus on Modeling A11 4. (a) 3, q 3 2 4 1 2 1 2, 1 (c) 3 − 5. (a) −4 3 1 15 −5 (b) (d) 1 1 q, 4 2, q 1 2 2 2 −4 3, q 2 3 4 (c) 2 x 1 and 2 x 3 (b) 2, 1, 2, 3 6. (a) 1194 13.93 y 7 2B 2 x 7 2B A (c) A 7. (a) (b) 2 97 2 y 5 (d) 13 13x 63 y 13 (b) C 5 x 69 5 3, 0, 3 y 2 1 (e) y 5 13x 10 x (bb) $18,750.00 1 2 2 25 (c) 18 years y 2x 6 8. (a) 9. 8 mi/h 10. (a) S 187.5x 2 x 3 1 FOCUS ON MODELING ■ page 198 1. (a) Regression line (b) y 1.8807x 82.65 (c) 191.7 cm y 180 170 160 150 ) 35 40 45 50 55 x Femur length (cm) 3. (a) y 100 80 60 40 20 ) r y ( e g A Regression line ANSWERS TO CUMULATIVE REVIEW TEST FOR CHAPTERS 1 AND 2 ■ page 190 9 5 i 12 3 1 2. (a) 3 (b) 16 (c) 1, 6 12 5 i (e) (b) 10 3i (d) (c) 2 3 1. $8000 3. (a) 2 (f) 1.75, 2.75 2 |
i 13 2 (d) 2, 2i 0 2 4 6 8 10 12 Diameter (in.) 14 16 x (b) y 6.451x 0.1523 (c) 116 years (b) y 4.857x 220.97 (c) 265 chirps/min 13. 1 subtract 1, take square root 0 15. x f (input) (output) 2 5 subtract 1, take square root subtract 1, take square root A12 Answers to Selected Exercises and Chapter Tests Regression line y 200 150 100 50 5. (a) 7. (a 50 60 70 80 Temperature (°F) 90 x y 25 20 15 10 5 0 Regression line 10 20 30 60 70 40 50 Flow rate (%) 80 90 100 x (b) y 0.168x 19.89 9. (a) y 80 (c) 8.13% 63. (b 75 70 65 60 55 Regression line 0 1920 1940 1960 1980 2000 x (b) y 0.2708x 462.9 11. (a) y 0.173x 64.717, y 0.269x 78.67 (b) 2045 (c) 80.3 years y 67. (a) (b 80 70 60 50 0 20 40 60 80 100 x Years since 1900 CHAPTER 3 SECTION 3.1 ■ page 211 1. value 3. (a) f and g (b) f x 5. 2 x 3 11. Subtract 4, then divide by 3 7 10 15 20 0 5 1 2 9. Square, then add 2 10 50 28.125 12.5 3.125 0 17. 3, 3, 6, 23 19. 3, 3, 2, 2a 1, 2a 1, 4, 94 21. 2a 2b 1 23. 4, 10, 2, x 2 2 0, 1, 0, 1 0 25. 6, 2, 1, 2, 29. 8, x 2 8x 16 3 4, undefined,, 1 3, 3, 2x 2 7x 1, 2x 2 3x 4 312 x2 1 27. 4, 1, 1, 2, 3 31. x 2 4x 5, x 2 6, 2 1 33. x 2 4, 35. 3a 2, 3 a h 1 2 2, 3 37. 5, 5, 0 39 41. 3 5a 4a 2, 3 |
5a 5h 4a 2 8ah 4h 2, 5 8a 4h q, q x x 1 5, q 1, 5 4 3 q, q 2 6, q 1 51. 6 3, q x 0 2, 3 5 55. 47. 61. 49. 57. 45. 43, q 5 B 4, q A B 65. (a) f x 1 53. 2 q, 0 2 3 4 59 10 3 0.08x.16 0.32 0.48 0.64 y 1 0 1 x (c) (c) y 2 0 2 x 2 10 1532.1, C 100 1 C 0 C 69. (a) 1 ing 10 yd and 100 yd is the volume of the full tank, and tank, 20 minutes later. (c) (c) 1 2 x 2100 2 1500 V 20 (b) The cost of produc71. (a) 50, 0 0 is the volume of the empty (b) V 2 1 1 2 Answers to Section 3.2 A13 0.1 √ 73. (a) central axis. (c) 1 r 0 0.1 0.2 0.3 0.4 0.5 4440, √ 0.4 1 2 2 1665 (b) Flow is faster near 9. √ r 2 1 4625 4440 3885 2960 1665 0 11. _5 y 2 0 _2 _2 2 x 13. 15. y 5 0 1 _4 x 17. 19 _5 21. 25. _5 3 x 2 x 23. _5 27. 5 x _5 75. (a) 8.66 m, 6.61 m, 4.36 m (b) It will appear to get shorter. 77. (a) $90, $105, $100, $105 (b) Total cost of an order, including shipping 40 x 1 2 if 0 x 40 if 40 x 65 if x 65 x 65 1 (c) Fines for violating the speed limits 2 t 79. (a) F x 1 2 • 15 0 15 (b) $150, $0, $150 81. T 0 83. Population (× 1000) P 900 850 800 750 700 1985 1990 1995 2000 t Years SECTION 3.2 ■ page 221, x3 2, 10, 10 3. 3 x f 1. 5. 1 2 y 4 0 −2 −4 7. 4 x |
_4 y 4 0 _4 29. (a) (b) 4 x −5 5 −5 5 −10 10 −10 y 2 0 _5 y 4 0 _4 y 1 0 y 5 0 _2 10 A14 Answers to Selected Exercises and Chapter Tests (c) −2 20 −5 10 Graph (c) is the most appropriate. 31. (a) (d) −10 (b) 100 −100 10 45. y 1 0 10 1 x 47. _7 49. f x 1 2 2 x 2 • if x 2 if 2 x 2 if x 2 7 _7 7 2 −3 3 51. (a) Yes domain 3 59. No 69. (a) 3, 2 4 61. No (b) No, range 3 (c) Yes (d) No 53. Function, 55. Not a function 57. Yes 2, 2 4 63. Yes 65. Yes 67. Yes (b) −2 (c) −3 2 −2 5 −10 3 Graph (c) is the most appropriate. 33. _5 37. y 2 0 _2 y 3 5 x (d) −10 35. _5 39. −10 10 −10 y 4 0 y 3 _3 0 3 x _3 41. 43. y 5 0 _5 5 x _5 _3 3 0 _2 y 5 0 x 5 x c=6 10 c=4 c=2 c=0 _5 5 _5 10 _10 c=0 c=_2 c=_4 c=_6 5 10 _10 x2 c (c) If c 0, then the graph of x f 1 graph of y x 2 shifted upward c units. If c 0, then the graph of x f 2 c units. 71. (a) is the same as the graph of y x 2 shifted downward is the same as the x2 c (b) 2 1 c=0 c=2 10 c=4 c=6 10 5 x _10 10 _10 10 _10 c=_6 c=_4 _10 c=_2 c=0 f (c) If c 0, then the graph of x is the same as the 2 1 graph of y x 3 shifted to the right c units. If c 0, then the graph is the same as the graph of y x 3 shifted to the x of 1 2 1 left c units. 73. (a) x c x |
c (b=1 2 c=1 4 c=1 6 4 3 2 c=1 c=1 3 c=1 5 3 2 1x. As c increases, the graph of (c) Graphs of even roots are similar to similar to near 0 and flatter when x 1. 7 75. 13 x 6 x 4 3 29 x2, 77. f x 1 2 ; graphs of odd roots are y c1x becomes steeper 79. 0.005 10 0 81. (a) E x 1 2 e (b) E (dollars) Answers to Section 3.3 A15 15. (a) 4.8 100 −4.75 4.75 −0.8 (b) Domain 3 range 3 0, 4 4 4, 4, 4 6 0.10x 36 0.06 x 300 1 0 x 300 x 300, 2 17. (a) 3 −1 −1 9 (b) Domain 0, q range 3 2 1, q 3, 2 27. (a) 2, 4, 4 3 3 5, 6 4 _5 10 0 100 x (kWh) 0.41 if 0 x 1 0.58 if 1 x 2 0.75 if 2 x 3 0.92 if 3 x 3.5 83. P x 1 2 P (dollars) 0.90 d 0.50 0.10 0 1 2 3 4 x (oz) 3. (a) increase, SECTION 3.3 page 232 1. a, 4 1, 2 5. (a) 1, 1, 3, 4 3, 2, 4 (d) (c) 4, 4 (b) Domain 3 9. (a) 4, 5, 4 3 3 3, 4 (b) Domain 3 3 x 2 and x 4, range 3 3 2, 3 4 4 4 4 (b) decrease, 1, 4, range 3 4 7. (a) 3, 2, 2, 1, 0 (b) Domain range 1 q, q q, q, 2 1 2 −3 3 11. (a) 13. (a) −6 −4 −3 6 −6 5 −12 6 4 (b) Domain range {4} 1 q, q, 2 (b) Domain 1 q, 4 range 1 4 q, q, 2 19. (a) 1, 1 3 3, 2, 4 (b) 3 23. (a) _ |
2 10 _10 2, 4, 3 1, 1 (b) 2, 2 4 21. (a) 2, 1 3 1, 2, 3 4 4 25. (a) 20 7 _3 5 _25 (b) Increasing on decreasing on 3 q, 2.5 2.5, q 1 ; 2 (b) Increasing on 2, q 2 3 29. (a) ; decreasing on 1, q, 1 4 1, 2 4 3 4 3 _3 5 _10 5 _5 10 0, 2 1, 3 2, q 2, 0 3 3, q q, 2 3 q, 0 ; decreasing on ; decreasing on (b) Increasing on 0.22, q ; 3 1.55, 0.22 (b) Increasing on decreasing on (b) Increasing on 0, 1 4 35. (a) Local maximum 0.38 (b) Increasing on q, 1.55, 4 1 decreasing on 3 31. (a) Local maximum 2 when x 0; local minimum 1 when x 2, local minimum 0 when x 2 2, 0 3 1 33. (a) Local maximum 0 when x 0; local maximum 1 when x 3, local minimum 2 when x 2, local minimum 1 when x 1 q, 2 2 1 when x 0.58; local minimum 0.38 when x 0.58 0.58, q (b) Increasing on 4 37. (a) Local maximum 0 when x 0; local 0.58, 0.58 4 3 minimum 13.61 when x 1.71, local minimum 73.32 when x 3.21 decreasing on 5.66 when x 4.00 on local minimum 0.38 when x 1.73 q, 1.73 1 4 43. (a) 500 MW, 725 MW (b) Between 3:00 A.M. and 4:00 A.M. (c) Just before noon 30, 32 decreasing on weight, only to regain it again later. 3.21, q 39. (a) Local maximum q, 4.00 ; decreasing 1 41. (a) Local maximum 0.38 when x 1.73; (b) He went on a crash diet and lost (b) Increasing on (b) Increasing on (b) Increasing on ; decreasing on ; decreasing on q, 1.71 q, 0.58 Increasing on 3 0, 3.21 1.73, 0 1.71, 0 4. |
00, 6.00 1.73, q 45. (a) 0, 1.73 32, 68 0, 30 ; ; A16 Answers to Selected Exercises and Chapter Tests 300, q 3 0, 150 (b) Local maximum when 47. (a) Increasing on 150, 300 3 when Runner B fell but got up again to finish second. 51. (a) 4 x 300 (b) Increases 2 x 150 ; local minimum 49. Runner A won the race. All runners finished. ; decreasing on 3 4 15. y 17. y 4 480 0 2 x 100 0 300 53. 20 mi/h 55. r 0.67 cm SECTION 3.4 page 240 1. 100 miles 2 hours 50 mi/h 3. 25 1 5 1 6 5. 2 3 7. 4 5 9. 3 11. 5 13. 60 15. 12 3h 1 a 17. 2 a h 25. (a) 245 persons/yr 19. a 1 2 1 2 23. 0.25 ft/day 21. (a) (b) 328.5 persons/yr 27. (a) 7.2 units/yr (d) 2000–2001, 2001–2002 next 20 minutes: 1.5°F/min; first interval (c) 1997–2001 (b) 8 units/yr (d) 2001–2006 (c) 55 units/yr 29. First 20 minutes: 4.05°F/min, (b) left 3. (a) x-axis SECTION 3.5 page 251 1. (a) up the left 2 units 2 units, then shift downward 2 units then shift upward 2 units 9. (a) (b) (b) Shift upward 2 units (b) y-axis 5. (a) Shift to 7. (a) Shift to the left (b) Shift to the right 2 unitsc) 11. y 1 0 y 1 0 (d) y 2 x 1 0 1 x 13 19. 21. y 5 0 y 1 0 1 23. 27. y 2 0 1 1 x 25. 29. x x 31. 33 1x 2 35. 39. 43 47. g 1 51. (a) 3 2 x2 b) 1 0 2 0 1 f 37. 1 2 2 2 (c) 2 2 x 2 41. f 45. 49. g (d 24 x 1 x 2 2 2 |
2 1x 2 1 53. (a) (b) 61. (a) 4 (b) Answers to Section 3.5 A17 1 3 For part (b) shrink the graph in (a) vertically by a factor of 1 ; for part (c) shrink the 3 graph in (a) vertically by a factor of and reflect in the x-axis; for part (d) shift the graph in (a) to the right 4 units, shrink vertically by a 1 factor of, and then reflect in 3 the x-axis. The graph in part (b) is shrunk horizontally by a factor of and the graph in part (c) is stretched by a factor of 2. 1 2 _4 6 (c) _4 (d) 63. _5 (b) (a) (c) 1 2 4 5 4 _4 65. Even 67. Neither y 1 0 1 x 69. Odd 71. Neither _5 73. (a) y 3 0 _3 y 3 5 x (b) y 2 −2 0 −2 2 x −2 0 2 x 75. To obtain the graph of g, reflect in the x-axis the part of the graph of f that is below the x-axisd) (f) (b _3 x 3 (c) (e) 55. (a) y 2 0 57. _3 59. (d) (c) _8 8 _2 (b) (a) 8 For part (b) shift the graph in (a) to the left 5 units; for part (c) shift the graph in (a) to the left 5 units and stretch vertically by a factor of 2; for part (d) shift the graph in (a) to the left 5 units, stretch vertically by a factor of 2, and then shift upward 4 units, x 1 2 ; A18 Answers to Selected Exercises and Chapter Tests 77. (a) (b) y 5 _5 0 _3 5 x _5 y 5 0 5 x 79. (a) Shift upward 4 units, shrink vertically by a factor of 0.01 t (b) Shift to the right 10 units; 4 0.01 t 10 2 g 1 2 1 2 3. Multiply by 2, then add 1; Add 1, then multiply q, q ; 2 1. 8, 2, 15, SECTION 3.6 page 260 3 |
5 f g x 2 x2 x 3, 1 q, q by 2 5. f g fg 2 1 x 1 1 2 x2 x 3, 1 q x3 3x2, x 3 2 x2 24 x2 21 x, 1 q, 0 x 2 24 x2 21 x, x 2 1 2x3 x2 4x 4, 0. 1 f g 1 fg., 4 x2 1 x 1 6x 8 x2 4x 1, 2 4, x 4, x 0 ; 1, 2 3 1, 2 ; 4 ; 4, x 4, x 0 ; 2 2x 8 x2 4x 8 x2 4x x 4 2x, x 4, x 0 ;, x 4, x 0 f g 2 1 x fg 15. y f+g g f 3 f f+g 19. _3 g _2 x 3 21. (a) 1 (b) 23 23. (a) 11 (b) 119 11. 0, 1 3 4 13. 3, q 1 2 17. _3 3 f+g g f 3 _1 1 _1 27. 4 29. 5 31. 4 4, x 0 ; 2 x q 2x x 1 1, 121x 161x, ; 2 0, q 3 0, q 2 3 8x 1, 1 2 2 1 f g x 25. (a) 3x 2 1 33 8x 11, 4x 9, 1 16x 5b) 9x 2 30x 23 q, q ; ; 2 1 q, q 2 1 q, q ; 2 q, q 2, 2 q, q ; 1 2 2 q, q ; 2 1 2 x 2 1 x2 1, x4, 1 x 2, q ; 2 q, q 2, x 2; 1 35 37. 1 f f 1 39 41. 1 g g 2 2x 4 2 1 x, x 0, 1 2 x 3 3, 1 q 4x 9, 0, x 1 0 0 ; q, q 2 2 x 2 1 q, q, 1 q, q ; 0 2 1 2x 1 2x 4x 12, ; 2 x 2 2 1 x 2x 1 4x 3, 121x x 2 2 1 19 x 43. 1 f f 1 45, q 0, q 3 q 2x 1 1 1x 5 x 51. 5 4 1 g x 1 2 x2, f x 1 |
2 x 53. 55. 49. 47. x 2 1 x 9, f 1 1 x 3, f x 0 1 x 1, f x2, g 1 1 4 x, f 13 x, g 2 0.15x 0.000002x2 1 pr2 r (b) f 1 x 65. (a) f f g 1 R 2 x 57. 59/x x 9 2 x 1 2 61. (a) 3600pt 2 x 100 x 1 f g t (c) 2 2 1 0.9x x (b) g 2 2 1 0.9x 90, g f (c) first rebate, then discount, g f: first discount, then rebate, g f is the better deal 0.9x 100 60t 2 1 63. 16pt2 SECTION 3.7 page 270 1. different, Horizontal Line 3. (a) Take the cube root, subtract 5, then divide the result by 3. (b) f x 1 2 1 3x 5 3, f 1 x 1 2 2 x1/3 5 3 5. No 7. Yes 9. No 2 x 39. 35. 11. Yes 13. Yes 15. No f1 x 23. 1 1 2x 14 x, x 4 x2 2x, x 1 x 1 1 21 2 f1 x 2 1 2x 3 / f1 f1 f1 f1 41. 5x 1 51. 47. 43 17. No 37. f1 x 2 1 2 f1 x 1 x 1 2 2 1/x 1 2 45. 2 f1 49. 53. f1 1 2 19. No 1 41 (b) 3 21. (a 51 x 4 x2 2 3 2 x 2 1 1 14 x Answers to Chapter 3 Review A19 55. (a) (b) _5 y 2 0 _5 5 x f y 5 0 _2 f–¡ 3 x _5 73. y 1 0 1 x (c) f1 57. (a) x 1 2 1 31 x 6 2 (b) y 2 f y 2 f–¡ _2 0 _1 2 x _2 0 _1 2 x f 1 (c) x 2 59. Not one-to-one 1 x2 1, x 0 _2 3 _3 61. One-to-one 2 _4 20 _20 16 500 80x x 75. |
(a) of hours worked as a function of the fee (b) f1 x f 2 1 1 x 500 1, the number 801 2 (c) 9; if he charges $1220, 2 he worked 9 h 77. (a) √1 t 1 2 B 0.25 t 18,500 (b) 0.498; F1 at a distance 0.498 from the central axis the velocity is 30 x 32 x 79. (a) 2 1 Fahrenheit temperature is x temperature is 86F, it is 30C ; the Celsius temperature when the 30 (b) ; when the 5 91 F1 86 2 2 1 81. (a) f x 1 2 0.1x 2000 0.2 e x 20,000 1 2 if 0 x 20,000 if x 20,000 (b) f1 x 1 2 10x 10,000 5x e if 0 x 2000 if x 2000 If you pay x euros in taxes, your income is f1 60,000 (c) x x dollars has 10,000 2 f 1 toppings. f1 83. x 1 2 1 1 21 f1 pizza costing 2 CHAPTER 3 REVIEW page 274 1. x2 5 f x 3. Add 10, then multiply the result by 3. 1 2 63. Not one-to-one 10 _5 15 _10 5 f1 x 1 2 x 2 65. (a) (b) _4 67. (a) (b) g1 x 1 2 g _4 f _1 4 4 f _4 x2 3, x 0 4 _4 4 −1 g 69. x 0, f1 x 1 2 24 x 71. x 2, h1 1x 1000 10,000 34,000, 205,000 (b) The costs of 7. (a) 5000 printing 1000 and 10,000 copies of the book (c) ; 0 fixed costs 9. 6, 2, 18, a 2 4a 6, a 2 4a 6, x 2 2x 3, 4x 2 8x 6, 2x 2 8x 10 (b) Function (c) Function, one-to-one 13. Domain 4, q 17. (d) Not a function q, q, range 3 x 2, 1, 0 11. (a) Not a function 3, q 19. 2 q, 1 0, q 1 21. 1, 4 15 23. 25. |
y 3 0 _3 _5 5 x _5 y 2 0 _5 5 t A20 Answers to Selected Exercises and Chapter Tests 29. y 1 0 _1 1 5 x 5 x 27. _5 31. y 5 0 (3, _3) _5 y 3 _3 y 1 0 _3 35. 39. _5 0 5 x y 2 _3 0 3 x _2 33. 37. 5 x 3 x _5 y 3 y 5 0 2 _2 55. 5 57. 3 1 3 h 1 2 59. (a) P 10 1 2 5010, P 20 1 2 7040 ; the 1 2 (b) Shift to the left (b) Yes, because it is a (c) Stretch vertically by a factor of 2, then shift upward (d) Shift to the right 2 units and downward 2 units populations in 1995 and 2005 (b) 203 people/yr; average annual 2, 1 population increase 61. (a) linear function 63. (a) Shift upward 8 units 8 units 1 unit (e) Reflect in y-axis (g) Reflect in x-axis (b) Odd (c) Even 71. Local maximum 3.79 when x 0.46; local minimum 2.81 when x 0.46 73. (f) Reflect in y-axis, then in x-axis (h) Reflect in line y x 1 (d) Neither 65. (a) Neither 7 69. 68 ft 67. g 1 2 10 f(x) 4 (f+g)(x) g(x) −4 −2 75. (a) (c) fg x 2 2 1 3x 2 x f g 1 x 2 1 f/ 9x 3x (d) (e) 1 77. 1 g f x 2 q, q 1 79 9x g g x 1 2 1 2 1 85. No 87. f 91. (a), (b) 1 2 x f g 2 6x 6 (b) 2 18x 8 4 3x x 3 13x 2 3x 2 / 1 2 15x 6 2 1 2 6x 1 ;, 2 1 f f q, q ; 2 2 1 1 3 6x 2 4x, 81. Yes 83. No 2 g f (f) 1 q, q 2 12x 3, 1 4 4x x x 2 1 1 |
1x x 2 3 89 13 x 1 x 2 2 x 2 2 1 3x 2 9x 2 9x 4, x 2 q, q 2 y 3 0 f _3 f –¡ _5 5 x 49. _10 _5 0 1 x 41. No 47. 43. Yes 45. (iii) 250 _20 2.1, 0.2 4 3 1.9, q 10 5 2 _30 3 51. 53. _2 6 _10 (c) f1 x 1 2 1x 4 10 CHAPTER 3 TEST page 278 1. (a) and (b) are graphs of functions, (a) is one-to-one a 1 0, q 2. (a) 2/3, 3. (a) x (b), x 2 16/5 1a/ 1 3 1, 0 (b) (c, 0 ; decreasing on, 1 4 Increasing on 2.67, q 3 0, 2.67 3 2 4 x f x 1 2 1 27 08 0 01 1 00 2 01 3 08 4 2 0 1 x (d) By the Horizontal Line Test; take the cube root, then add 2 4000 ; total 4 1 2 12. (a) 20 (b) No Answers to Section 4.1 A21 1 x f1 x1/3 2 (e) sales revenue with prices of $2 and $4 (b) 4. (a) 5000 R 2 2 2 1 4000, R revenue increases until price reaches $3, then decreases 0 5 (c) $4500; $3 5. 5 6. (a) (b) y 3 0 _3 _5 5 x _5 y 3 0 _3 7. (a) Shift to the right 3 units, then shift upward 2 units (b) Reflect in y-axis 8. (a) 3, 0 (b) y 3 0 1 x _4 4 _30 (c) Local minimum 27.18 when x 1.61; local maximum 2.55 when x 0.18; local minimum 11.93 when x 1.43 (d) ; decreasing on (e) Increasing on 3 0.18, 1.43 q, 1.61 1.61, 0.18 27.18, q 1.43 FOCUS ON MODELING page 285 5 x 1. 5, „ 0 3„ 10x x 1A/p, A |
0 25t, t 0 V 3. „ 2 2, 0 x 10 1 1 2„ 3, „ 0 2, x 0 1 x 13/4 x A 7. 2 1 2 x 2 240/x, x 0 x 2 b14 b, 0 b 4 2 11. S 1 15. A b 9. r 13. 2 1 h 17. A 1 19. (b) 2 p 21. (b) 23. (a) 2 length is 40 ft x 2h2100 h 19 x x 1 2400 2 x x 2 8„ 7200/„, 0 h 10 (c) 9.5, 9.5 (c) 600 ft by 1200 ft (b) Width along road is 30 ft, A 25. (a) (c) 15 ft to 60 ft p 4 8 (b) Width 8.40 ft, height of rectangular part 4.20 ft 27. (a) 15x b) Height 1.44 ft, width 2.88 ft 1 2 x 2 48/x 2x 200/x 29. (a) A x 1 2 (b) 10 m by 10 m 9. (a) (c) 2 10. (a) (b) f _5 2 1 f g 1 (d) 2 f1 e) x 2 3 x2b) 1 x 9 x2 2 x 2 2 1 31. (b) To point C, 5.1 mi from B CHAPTER 4 y 3 0 _3 5 x f –¡ 2 1, 3 3, 5 SECTION 4.1 page 297 1. square 3. upward, 1 q, 4, 7. (a) (c) 4 1 9. (a) x f 1 2 3, 9 (b) Vertex x-intercepts 0, 6 y-intercept 0 (c, minimum 5. (a) (b) 3 f (c) 2 2 11. (a) x A 2 1 2, 3 (b) Vertex A x-intercepts 0, 3, y-intercept 0 (c) (b) 4 3, 4 2 3, q 1, 3 x 3 2B 9 2B 2 2 9 2 11. (a) Domain 0, 6 3 (b) y, range 3 4 1, 7 4 (c) 5 4 y 3 −3 0 3 x −3 y 2 −2 0 2 x 1 0 1 x |
A22 Answers to Selected Exercises and Chapter Tests x 2 2 1 2 f 13. (a) x 1 (b) Vertex (c) 1 2 2, 1 1 y ; x-intercepts 1, 3; y-intercept 3 2 2 −2 0 2 x f 15. (a) x 1 (b) Vertex (c) 2 3, 13 1 y 2 2 13 x 3 2 1 ; x-intercepts 3 113 ; y-intercept 4 6 −2 2 x 2 1 2 1, 1 2 1 x 1 ; no x-intercept; y-intercept 3 2 x f 17. (a) 1 (b) Vertex (c) 1 2 y 3 23. (a) (bc) Minimum f 1 1 2 2 3 0 _3 _2 (_1, _2) x 2 25. (a) (bc) Minimum f 2 1 1 2 2 0 1 (1, −2) x 27. (a) (b) f x 1 2 x 3 2B y A 2 21 4 21 3!_, @ 4 2 (c) Maximum f 3 2B A 21 4 3 0 _2 _3 x 3 _3 0 3 x 29. (a) (bc) Minimum g 1 2 1 2 f 19. (a) x 1 (b) Vertex (c) y 2 2 5, 7 1 2 2 7 x 5 1 ; no x-intercept; y-intercept 57 2 7 0 _2 5 x 4 2 2, 19 x 2 2 19 2 1 ; x-intercepts x f 21. (a) 1 (b) Vertex (c) 1 2 1 2 119 ; y-intercept 3 2 y 5 _2 0 1 x 10 0 (2, 1) 6 x 31. (a) (b) h x 1 2 x 1 2B y A 2 5 4 (c) Maximum h 1 2B A 5 4 1!_, @ 2 5 4 2 _4 0 _2 2 x 35. Maximum 1 33. Minimum f 2B A 37. Minimum f 0.6 1 2 1 41. Maximum f 2 1 q, 1 q, q 45., 3 4 15.64 7 43. 2 1 2 1 47. 1 4 1 f 3.5 h 1 2x2 4x 2, q 39. Minimum x f 2 q, q 23, 1 2 3 B 185 |
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