2. Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 12-1The Counting Principle Lesson 12-2Permutations and Combinations Lesson 12-3Probability Lesson 12-4Multiplying Probabilities Lesson 12-5Adding Probabilities Lesson 12-6Statistical Measures Lesson 12-7The Normal Distribution Lesson 12-8Binomial Experiments Lesson 12-9Sampling and Error

5. Lesson 1 Contents Example 1Independent Events Example 2Fundamental Counting Principle Example 3More than Two Independent Events Example 4Dependent Events

6. Example 1-1a A sandwich menu offers customers a choice of white, wheat, or rye bread with one spread chosen from butter, mustard, or mayonnaise. How many different combinations of bread and spread are there? First, note that the choice of the type of bread does not affect the choice of the type of spread, so these events are independent. Method 1Tree Diagram Let W represent white, H, wheat, R, rye, B, butter, M, mustard, and Y, mayonnaise. Make a tree diagram in which the first row shows the choice of bread and the second row shows the choice of spread.

7. Example 1-1a Answer: There are nine possible outcomes. Bread Spread Possible Combinations

8. Example 1-1a Answer: This method also shows that there are nine outcomes. Method 2Make a Table Make a table in which each row represents a type of bread and each column represents a type of spread. Spread ButterMustardMayo Bread WhiteWBWMWY WheatHBHMHY RyeRBRMRY

9. Example 1-1b A pizza place offers customers a choice of American, mozzarella, Swiss, feta, or provolone cheese with one topping chosen from pepperoni, mushrooms or sausage. How many different combinations of cheese and toppings are there? Answer: 15

10. Example 1-2a Multiple-Choice Test Item For their vacation, the Murray family is choosing a trip to the beach or to the mountains. They can select their transportation from a car, plane, or train. How many different ways can they select a destination followed by a means of transportation? A 2 B 5 C 6 D 9 Read the Test Item Their choice of destination does not affect their choice of transportation, so these events are independent.

11. Example 1-2a Solve the Test Item There are two ways they can choose a destination and 3 ways they can choose a means of transportation. By the Fundamental Counting Principle, there are 2 3 or 6 total ways to choose. Answer: C

12. Example 1-2b Multiple-Choice Test Item For their vacation, the Esper family is going on a trip. They can select their transportation from a car, plane, or train. They can also select from 4 different hotels. How many different ways can they select a means of transportation followed by a hotel? A 8 B 12 C 16 D 7 Answer: B

13. Example 1-3a Communication How many codes are possible if an answering machine requires a 2-digit code to retrieve messages? The choice of any digit does not affect the other digit, so the choices of digits are independent events. There are 10 possible choices for the first digit and 10 possible choices for the second digit. Answer: So, there are or 100 possible different codes.

14. Example 1-3b Banking Many automated teller machines (ATM) require a 4-digit code to access an account. How many codes are possible? Answer: 10,000

15. Example 1-4a How many different schedules could a student have who is planning to take 4 different classes? Assume each class is offered each period. When a student schedules a given class for a given period, he or she cannot schedule that class for any other period. Therefore, the choices of which class to schedule each period are dependent events. There are 4 classes that can be taken during the first period. That leaves 3 classes for the second period, 2 classes for the third period, and so on.

16. Example 1-4a Period1st2nd3rd4th Number of Choices4321 Answer: There areor 24 different schedules for a student who is taking 4 classes.

17. Example 1-4b Answer: 120 How many different schedules could a student have who is planning to take 5 different classes?

18. End of Lesson 1

19. Lesson 2 Contents Example 1Permutation Example 2Permutation with Repetition Example 3Combination Example 4Multiple Events

20. Example 2-1a Eight people enter the Best Pic contest. How many ways can blue, red, and green ribbons be awarded? Since each winner will receive a different ribbon, order is important. You must find the number of permutations of 8 things taken 3 at a time. Permutation formula Simplify.

21. Example 2-1a Divide by common factors. 11111 11111 Answer: The ribbons can be awarded in 336 ways.

22. Example 2-1b Ten people are competing in a swim race where 4 ribbons will be given. How many ways can blue, red, green, and yellow ribbons be awarded? Answer: 5040

23. Example 2-2a How many different ways can the letters of the word BANANA be arranged? The second, fourth, and sixth letters are each A. The third and fifth letters are each N. You need to find the number of permutations of 6 letters of which 3 of one letter and 2 of another letter are the same. Answer: There are 60 ways to arrange the letters.

24. Example 2-2b How many different ways can the letters of the word ALGEBRA be arranged? Answer: 2520

25. Example 2-3a Five cousins at a family reunion decide that three of them will go to pick up a pizza. How many ways can they choose three people to go? Since the order they choose is not important, you must find the number of combinations of 5 cousins taken three at a time. Combination formula

26. Example 2-3a Answer: There are 10 ways to choose three people from the five cousins. Simplify.

27. Example 2-3b Six friends at a party decide that three of them will go to pick up a movie. How many ways can they choose three people to go? Answer: 20 ways

28. Example 2-4a Six cards are drawn from a standard deck of cards. How many hands consist of two hearts and four spades? By the Fundamental Counting Principle, you can multiply the number of ways to select two hearts and the number of ways to select four spades. Only the cards in the hand matter, not the order in which they were drawn, so use combinations. C(13, 2) Two of 13 hearts are to be drawn. C(13, 4) Four of 13 spades are to be drawn.

29. Example 2-4a Answer: There are 55,770 hands consisting of 2 hearts and 4 spades. Combination formula Subtract. Simplify.

30. Example 2-4b Thirteen cards are drawn from a standard deck of cards. How many hands consist of six hearts and seven diamonds? Answer: 2,944,656 hands

31. End of Lesson 2

32. Lesson 3 Contents Example 1Probability Example 2Probability with Combinations Example 3Odds Example 4Probability Distribution

33. Example 3-1a When three coins are tossed, what is the probability that all three are heads? You can use a tree diagram to find the sample space. First Coin Second Coin Third Coin Possible Outcomes

34. Example 3-1a Answer: The probability of tossing three heads is Probability formula There are 8 possible outcomes. You can confirm this using the Fundamental Counting Principle. There are 2 possible results for the first coin and 2 for the second coin, so there areor 8 outcomes. Only one of these outcomes is HHH so The other outcomes are failures, so

35. Example 3-1b When three coins are tossed, what is the probability that exactly two are heads? Answer:

36. Example 3-2a Roman has a collection of 26 books–16 are fiction and 10 are nonfiction. He randomly chooses 8 books to take with him on vacation. What is the probability that he chooses 4 fiction and 4 nonfiction? Step 1Determine how many 8-book selections meet the conditions. C(16, 4) Select 4 fiction books. Their order does not matter. C(10, 4) Select 4 nonfiction.

37. Example 3-2a Step 2Use the Fundamental Counting Principle to find the number of successes. Step 3Find the total number, s + f, of possible 8-book selections.

38. Example 3-2a Step 4Determine the probability. P(4 fiction, 4 nonfiction ) Probability formula Substitute. Use a calculator. Answer: The probability is about 0.24464 or 24.5%.

39. Example 3-2b Ainsley has a collection of 15 CDs–5 are jazz and 10 are blues. She randomly chooses 7 CDs to take with her on vacation. What is the probability that she chooses 2 jazz and 5 blues? Answer: The probability is about 0.392 or 39.2%.

40. Example 3-3a Life Expectancy According to the U.S. National Center for Health Statistics, the chances of a male born in 1990 living to be at least 65 years of age are about 3 in 4. For females, the chances are about 17 in 20. What are the odds that a male born in 1990 will die before age 65? Three out of four males will live to be at least 65, so the number of successes (living to 65) is 3. The number of failures is

41. Example 3-3a Odds formula Answer: The odds of a male dying before age 65 are 1:3.

42. Example 3-3a What are the odds that a female born in 1990 will die before age 65? Seventeen out of twenty females will live to be at least 65, so the number of successes in this case is 17. The number of failures is Answer: The odds of a female dying before age 65 are 3:17. Odds formula

43. Life Expectancy The chances of a male born in 1980 to live to be at least 65 years of age are about 7 in 10. For females, the chances are about 21 in 25. a. What are the odds that a male born in 1980 will live to age 65? b. What are the odds that a female born in 1980 will live to age 65? Example 3-3b Answer: 7:3 Answer: 21:4

44. Example 3-4a Suppose two dice are rolled. The table and the relative-frequency histogram show the distribution of the sum of the numbers rolled. Probability 12111098765432S = Sum

45. Example 3-4a Use the graph to determine which outcomes are least likely. What are their probabilities? Answer: The least probability is The least likely outcomes are sums of 2 and 12.

46. Example 3-4a Use the table to find What other sum with this probability is a sum of 3? Probability 12111098765432S = Sum Answer:The other sum with this probability is a sum of 3.

47. Example 3-4a What are the odds of rolling a sum of 5? Step 1 Identify s and f. Step 2 Find the odds. Answer:So, the odds of rolling a sum of 5 are 1:8.

48. Example 3-4b Suppose two dice are rolled. The table and the relative frequency histogram show the distribution of the sum of the numbers rolled. Probability 12111098765432S = Sum

49. Example 3-4b a. Use the graph to determine which outcomes are the second most likely. What are their probabilities? Answer:

50. Probability 12111098765432S = Sum Example 3-4b b. Use the table to findWhat other sum has the same probability? Answer:

51. Example 3-4b c. What are the odds of rolling a sum of 3? Answer:1:18 Probability 12111098765432S = Sum

52. End of Lesson 3

53. Lesson 4 Contents Example 1Two Independent Events Example 2Three Independent Events Example 3Two Dependent Events Example 4Three Dependent Events

54. Example 4-1a Gernardo has 9 dimes and 7 pennies in his pocket. He randomly selects one coin, looks at it, and replaces it. He then randomly selects another coin. What is the probability that both of the coins he selects are dimes? ExploreThese two events are independent since he replaces the coin. The second event is not affected by the first. PlanSince there are 16 coins, the probability of getting a dime is

55. Example 4-1a Solve Probability of independent events Substitute and multiply. Answer: The probability that both coins are dimes isor about 31.6 %.

56. Example 4-1a Examine You can verify this by making a tree diagram that includes probabilities. Let D stand for dimes and P stand for pennies.

57. Example 4-1b Gernardo has 9 dimes and 7 pennies in his pocket. He randomly selects one coin, looks at it, and replaces it. He then randomly selects another coin. What is the probability that both of the coins he selects are pennies? Answer:or about 0.19

58. Example 4-2a When three dice are rolled, what is the probability that two dice show a 5 and the third die shows an even number? Let A be the event that the first die is a 5. Let B be the event that the second die is a 5. Let C be the event that the third die is even.

59. Example 4-2a Substitute and multiply. Answer: The probability that two dice show a five and the third die shows an even number is Probability of independent events

60. Example 4-2b When three dice are rolled, what is the probability that one die is a multiple of 3, one die shows an even number, and one die shows a 5? Answer:

61. Example 4-3a The host of a game show draws chips from a bag to determine the prizes for which contestants will play. Of the 20 chips in the bag, 11 show computer, 8 show trip, and 1 shows truck. If the host draws the chips at random and does not replace them, find each probability.

62. Example 4-3a a computer, then a truck Dependent events After the first chip is drawn, there are 19 left. Answer: The probability of a computer then a trip isor about 0.03.

63. Example 4-3a two trips Dependent events If the first chip shows trip, then 7 of the remaining 19 show trip. Answer: The probability of the host drawing two trips is

64. The host of a game show draws chips from a bag to determine the prizes for which contestants will play. Of the 20 chips in the bag, 11 show computer, 8 show trip, and 1 shows truck. If the host draws the chips at random and does not replace them, find each probability. a. a truck, then a trip b. two computers Example 4-3b Answer:

65. Example 4-4a Three cards are drawn from a standard deck of cards without replacement. Find the probability of drawing a heart, another heart, and a spade in that order. If the first two cards are hearts, then 13 of the remaining cards are spades. Answer: The probability isor about 0.015.

66. Example 4-4b Three cards are drawn from a standard deck of cards without replacement. Find the probability of drawing a diamond, another diamond, and another diamond in that order. Answer:

67. End of Lesson 4

68. Lesson 5 Contents Example 1Two Mutually Exclusive Events Example 2Three Mutually Exclusive Events Example 3Inclusive Events

69. Example 5-1a Sylvia has a stack of playing cards consisting of 10 hearts, 8 spades, and 7 clubs. If she selects a card at random from this stack, what is the probability that it is a heart or a club? These are mutually exclusive events since the card cannot be both a heart and a club. Note there is a total of 25 cards. Mutually exclusive events Substitute and add.

70. Example 5-1a Answer: The probability that Sylvia selects a heart or a club is

71. Example 5-1b Sylvia has a stack of playing cards consisting of 10 hearts, 8 spades, and 7 clubs. If she selects a card at random from this stack, what is the probability that it is a spade or a club? Answer:

72. Example 5-2a The Film Club makes a list of 9 comedies and 5 adventure movies they want to see. They plan to select 4 titles at random to show this semester. What is the probability that at least two of the films they select are comedies? At least 2 comedies means that the selected movies may include 2, 3 or 4 comedies. It is not possible to select 2 comedies, 3 comedies, and 4 comedies all in the same semester, so the events are mutually exclusive. Add the probabilities of each type of movie.

73. Example 5-2a Simplify. P(2) 2 comedies, 2 adventures P(3) 3 comedies, 1 adventure P(4) 4 comedies, 0 adventures Answer: The probability that at least two of the films are comedies isor about 0.91.

74. Example 5-2b The Book Club makes a list of 9 mysteries and 3 romance books they want to read. They plan to select 3 titles at random to read this semester. What is the probability that at least two of the books they select are romances? Answer:or about 0.04

75. Example 5-3a There are 2400 subscribers to an Internet service provider. Of these, 1200 own Brand A computers, 500 own Brand B, and 100 own both A and B. What is the probability that a subscriber selected at random owns either Brand A or Brand B? Since some subscribers own both A and B, the events are inclusive. P(A)P(A)P(B)P(B)

76. P( A or B ) = P( A ) + P( B ) – P( A and B ) Example 5-3a Substitute and simplify. Answer: The probability that a subscriber owns either A or B is

77. Example 5-3b There are 200 students taking Calculus, 500 taking Spanish, and 100 taking both. There are 1000 students in the school. What is the probability that a student selected at random is taking Calculus or Spanish? Answer:

78. End of Lesson 5

79. Lesson 6 Contents Example 1Choose a Measure of Central Tendency Example 2Standard Deviation

80. Example 6-1a A new Internet company has 3 employees who are paid $300,000, 10 who are paid $100,000, and 60 who are paid $50,000. Which measure of central tendency best represents the pay at this company? Since most of the employees are paid $50,000, the higher values are outliers. Answer: Thus, the median or mode best represents the available jobs.

81. Example 6-1a A new Internet company has 3 employees who are paid $300,000, 10 who are paid $100,000, and 60 who are paid $50,000. Which measure of central tendency would recruiters for this company most likely use to attract job applicants? Answer: The recruiters would use the mean ($67,123) to make it look like the employees would make more money.

82. Example 6-1b In a cereal contest, there is 1 Grand Prize of $1,000,000, 10 first prizes of $100, and 50 second prizes of $10. a. Which measure of central tendency best represents the prizes? b. Which measure of central tendency would advertisers be most likely to use? Answer: median or mode Answer: mean

83. Example 6-2a Rivers This table shows the length in thousands of miles of some of the longest rivers in the world. Find the standard deviation for these data. River Length (thousands of miles) Nile4.16 Amazon4.08 Missouri2.35 Rio Grande1.90 Danube1.78

84. Example 6-2a Step 1 Find the mean. Add the data and divide by the number of items. The mean is about 2.85 thousand miles.

85. Example 6-2a Step 2 Find the variance. Variance formula 5 Simplify. The variance is about 1.104 thousand miles or 1,104 miles.

86. Example 6-2a Step 3 Find the standard deviation. Take the square root of each side. Answer: The standard deviation is about 1.05 thousand miles.

87. Example 6-2b A teacher has the following test scores: 100, 4, 76, 85, and 92. Find the standard deviation for these data. Answer: 34.62

88. End of Lesson 6

89. Lesson 7 Contents Example 1Classify a Data Distribution Example 2Normal Distribution

90. Example 7-1a Determine whether the data {31, 33, 37, 35, 33, 36, 34, 36, 32, 36, 33, 32, 34, 34, 35, 34} appear to be positively skewed, negatively skewed, or normally distributed. Make a frequency table for the data. Then use the table to make a histogram. Value313233343536 Frequency123423

91. Example 7-1a Answer: Since the data are somewhat symmetric, this is a normal distribution.

92. Example 7-1b Determine whether the data {7, 5, 6, 7, 8, 4, 6, 8, 7, 6, 6, 4} appear to be positively skewed, negatively skewed, or normally distributed. Answer: negatively skewed

93. Example 7-2a Students counted the number of candies in 100 small packages. They found that the number of candies per package was normally distributed with a mean of 23 candies per package and a standard deviation of 1 piece of candy. About how many packages had between 24 and 22 candies? Draw a normal curve. Label the mean and positive and negative multiples of the standard deviation.

94. Example 7-2a The values of 22 and 24 are 1 standard deviation below and above the mean, respectively. Therefore, 68% of the data are located here. Multiply 100 by 0.68. Answer: About 68 packages contained between 22 and 24 pieces.

95. Example 7-2a What is the probability that a package selected at random had more than 25 candies? The value 25 is two standard deviations above the mean. You know that about 100% – 95% or 5% of the data are more than two standard deviations away from the mean. By the symmetry of the normal curve, half of 5%, or 2.5%, of the data are more than two standard deviations above the mean. Answer: The probability that a package selected at random had more than 25 candies is about 2.5% or 0.025.

96. Example 7-2b Students counted the number of candies in 100 small packages. They found that the number of candies per package was normally distributed with a mean of 23 candies per package and a standard deviation of 1 piece of candy. a. About how many packages had between 25 and 21 candies? b.What is the probability that a package selected at random had greater than 24 candies? Answer: 95 Answer: 16% or 0.16

97. End of Lesson 7

98. Lesson 8 Contents Example 1Binomial Theorem Example 2Binomial Experiment

99. Example 8-1a If a family has 4 children, what is the probability that they have 2 girls and 2 boys? There are two possible outcomes for the gender of each of their children: boy or girl. The probability of a boy and the probability of a girl The term represents 2 girls and 2 boys.

100. Example 8-1a Multiply. Answer: The probability of 2 boys and 2 girls is

101. Example 8-1b If a family has 4 children, what is the probability that they have 4 boys? Answer: The probability of 4 boys is.

102. Example 8-2a A report said that approximately 1 out of 6 cars sold in a certain year was green. Suppose a salesperson sells 7 cars per week. What is the probability that this salesperson will sell exactly 3 green cars in a week? The probability that a sold car is green is The probability that a sold car is not green is There are C(7, 3) ways to choose the three green cars that sell.

103. Answer: The probability that he will sell at least 3 green cars is or about 0.096. Example 8-2a If he sells three green cars, he sells four that are not green. Simplify.

104. Example 8-2b A report said that approximately 1 out of 6 cars sold in a certain year was green. Suppose a salesperson sells 7 cars per week. a. What is the probability that this salesperson will sell exactly 4 green cars in a week? b. What is the probability that this salesperson will sell at least 2 green cars in a week? Answer: 0.016 Answer: 0.33

105. End of Lesson 8

106. Lesson 9 Contents Example 1Biased and Unbiased Samples Example 2Find a Margin of Error Example 3Analyze a Margin of Error

107. Example 9-1a State whether the following method would produce a random sample. Explain. surveying people going into an action movie to find out the most popular kind of movie Answer: No; they will most likely think that action movies are the most popular kind of movie.

108. Example 9-1a State whether the following method would produce a random sample. Explain. calling every 10 th person on the list of subscribers to the newspaper to ask about the quality of service Answer: Yes; no obvious bias exists in calling every 10 th caller.

109. Example 9-1b State whether each method would produce a random sample. Explain. a.surveying people going into a football game to find out the most popular sport b.surveying every fifth person going into a mall to find out the most popular kind of movie Answer: No; they will most likely think that football is the most popular kind of sport. Answer: Yes; no obvious bias exists in asking every 5 th person.

110. Example 9-2a In a survey of 100 randomly selected adults, 37% answered “yes” to a particular question. What is the margin of error? Formula for margin of sampling error Use a calculator.

111. Example 9-2a Answer:The margin of error is about 10%. This means that there is a 95% chance that the percent of people in the whole population who would answer “yes” is between and

112. Example 9-2b In a survey of 100 randomly selected adults, 50% answered “no” to a particular question. What is the margin of error? Answer: 10%

113. Example 9-3a Health In an earlier survey, 30% of the people surveyed said they had smoked cigarettes in the past week. The margin of error was 2%. What does the 2% indicate about the results? Answer:There is a 95% chance that the percent of people in the population who had smoked cigarettes in the past week was between 28% and 32%.

114. Example 9-3a Health In an earlier survey, 30% of the people surveyed said they had smoked cigarettes in the past week. The margin of error was 2%. How many people were surveyed? Formula for margin of sampling error Divide each side by 2.

115. Example 9-3a Answer: About 2100 people were surveyed. Square each side. Multiply by n and divide by 0.0001. Use a calculator.

116. Example 9-3b Health In an earlier survey, 25% of the people surveyed said they had exercised in the past week. The margin of error was 2%. a. What does the 2% indicate about the results? b. How many people were surveyed? Answer: There is a 95% chance that the percent of people in the population that had exercised in the past week was between 23% and 27%. Answer: 1875

117. End of Lesson 9

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