1. Slide 9- 1

2. Chapter 9 Discrete Mathematics

3. 9.1 Basic Combinatorics

4. Slide 9- 4 Quick Review

5. Slide 9- 5 What you’ll learn about Discrete Versus Continuous The Importance of Counting The Multiplication Principle of Counting Permutations Combinations Subsets of an n-Set … and why Counting large sets is easy if you know the correct formula.

6. Slide 9- 6 Multiplication Principle of Counting

7. Slide 9- 7 Example Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. You can fill in the first blank 26 ways, the second blank 26 ways, the third blank 26 ways, the fourth blank 26 ways, the fifth blank 10 ways, the sixth blank 10 ways, and the seventh blank 10 ways. By the Multiplication Principle, there are 26×26×26×26×10×10×10 = 456,976,000 possible license plates.

8. Slide 9- 8 Permutations of an n-Set There are n! permutations of an n-set.

9. Slide 9- 9 Example Distinguishable Permutations Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER. Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations.

10. Slide 9- 10 Distinguishable Permutations

11. Slide 9- 11 Permutations Counting Formula

12. Slide 9- 12 Combination Counting Formula

13. Slide 9- 13 Example Counting Combinations How many 10 person committees can be formed from a group of 20 people?

14. Slide 9- 14 Formula for Counting Subsets of an n-Set

15. 9.2 The Binomial Theorem

16. Slide 9- 16 Quick Review

17. Slide 9- 17 What you’ll learn about Powers of Binomials Pascal’s Triangle The Binomial Theorem Factorial Identities … and why The Binomial Theorem is a marvelous study in combinatorial patterns.

18. Slide 9- 18 Binomial Coefficient

19. Slide 9- 19 Example Using n C r to Expand a Binomial

20. Slide 9- 20 Recursion Formula for Pascal’s Triangle

21. Slide 9- 21 The Binomial Theorem

22. Slide 9- 22 Basic Factorial Identities

23. 9.3 Probability

24. Slide 9- 24 Quick Review Solutions

25. Slide 9- 25 What you’ll learn about Sample Spaces and Probability Functions Determining Probabilities Venn Diagrams and Tree Diagrams Conditional Probability Binomial Distributions … and why Everyone should know how mathematical the “laws of chance” really are.

26. Slide 9- 26 Probability of an Event (Equally Likely Outcomes)

27. Slide 9- 27 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Possible outcomes for two rolls of a die

28. Slide 9- 28 1.Find the probability that the sum is a 2 2.Find the probability that the sum is a 3 3.Find the probability that the sum is a 4 4.Find the probability that the sum is a 5 5.Find the probability that the sum is a 6 6.Find the probability that the sum is a 7 7.Find the probability that the sum is a 8 8.Find the probability that the sum is a 9 9.Find the probability that the sum is a 10 10. Find the probability that the sum is a 11 11.Find the probability that the sum is a 12 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/26 3/36 2/36 1/36 Find the following probabilities

29. Slide 9- 29 Find the following probabilities when rolling two dice 1.P(sum is less than 4) 2.P(sum is even) 3.P(sum is odd or greater than 10) 4.P(doubles) 3/36 = 1/12 18/36 = 1/2 19/36 6/36 = 1/6

30. Slide 9- 30 Example Rolling the Dice Find the probability of rolling a sum divisible by 4 on a single roll of two fair dice.

31. Slide 9- 31 Probability Function

32. Slide 9- 32 Probability of an Event (Outcomes not Equally Likely)

33. Slide 9- 33 Strategy for Determining Probabilities

34. Slide 9- 34 Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

35. Slide 9- 35 Multiplication Principle of Probability Suppose an event A has probability p 1 and an event B has probability p 2 under the assumption that A occurs. Then the probability that both A and B occur is p 1 p 2.

36. Slide 9- 36 Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

37. Slide 9- 37 Example Choosing Chocolates Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

38. Slide 9- 38 Conditional Probability Formula

39. Slide 9- 39 Binomial Theorem Features of a Binomial Experiment 1.There are a fixed number of trials, denoted by the letter n. 2.The n trials are independent and repeated under identical conditions. 3.Each trial has only two outcomes, success denoted by p or failure denoted by q. 4.For each individual trial, the probability of success is the same. 5.The central problem in a binomial experiment to to find the probability of r successes in n trials

40. Slide 9- 40 n = fixed number of trials r = specific number of successes in n trials p = probability of success in one of n trials q = probability of failure in one of n trials ( q = 1 - p ) P(r)= probability of getting exactly r success among n trials Be sure that r and p both refer to the same category being called a success. Binomial Theorem

41. Slide 9- 41  P(r) = p r q n-r ( n - r ) ! r ! n !n !  P(r) = n C r p r q n-r for calculators with n C r key. Binomial Theorem

42. Slide 9- 42 Binomial Theorem This is a binomial experiment where: n = 5 r = 3 p = 0.90 q = 0.10 Example : Find the probability of getting exactly 3 correct responses among 5 different requests from AT&T directory assistance. Assume in general, AT&T is correct 90% of the time.

43. Slide 9- 43 Binomial Theorem n = 5 r = 3 p = 0.90 q = 0.10 Using the binomial probability formula to solve: P(3) = 5 C 3 0.9 3 0.1 2 =.0729

44. Slide 9- 44 Using n = 5 and p = 0.90, find the following: a) The probability of exactly 3 successes b) The probability of at least 3 successes a) P(3) = 0.073 b) P(at least 3) = P(3 or 4 or 5) = P(3) or P(4) or P(5) = 0.073 + 0.328 + 0.590 = 0.991 Binomial Theorem

45. Slide 9- 45 Binomial Theorem P(2 Green) P(1 Green) P(0 Green) There are 3 red balls and two green balls in a bag. Find the probability without replacement of drawing two balls with the following results:

46. Slide 9- 46 Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?

47. Slide 9- 47 Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?

48. Slide 9- 48 Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?

49. Slide 9- 49 Example Shooting Free Throws Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?

50. 9.4 Sequences

51. Slide 9- 51 Quick Review

52. Slide 9- 52 What you’ll learn about Infinite Sequences Limits of Infinite Sequences Arithmetic and Geometric Sequences Sequences and Graphing Calculators … and why Infinite sequences, especially those with finite limits, are involved in some key concepts of calculus.

53. Slide 9- 53 Limit of a Sequence

54. Slide 9- 54 Example Finding Limits of Sequences

55. Slide 9- 55 Arithmetic Sequence

56. Slide 9- 56 Example Arithmetic Sequences Find (a) the common difference, (b) the tenth term, (c) recursive rule for the nth term, and (d) an explicit rule for the nth term. -2, 1, 4, 7, …

57. Slide 9- 57 Geometric Sequence

58. Slide 9- 58 Example Defining Geometric Sequences Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. 2, 6, 18,…

59. Slide 9- 59 Sequences and Graphing Calculators One way to graph a explicitly defined sequences is as scatter plots of the points of the form (k,a k ). A second way is to use the sequence mode on a graphing calculator.

60. Slide 9- 60 The Fibonacci Sequence

61. 9.5 Series

62. Slide 9- 62 Quick Review

63. Slide 9- 63 What you’ll learn about Summation Notation Sums of Arithmetic and Geometric Sequences Infinite Series Convergences of Geometric Series … and why Infinite series are at the heart of integral calculus.

64. Slide 9- 64 Summation Notation

65. Slide 9- 65 Sum of a Finite Arithmetic Sequence

66. Slide 9- 66 Example Summing the Terms of an Arithmetic Sequence

67. Slide 9- 67 Sum of a Finite Geometric Sequence

68. Slide 9- 68 Infinite Series

69. Slide 9- 69 Sum of an Infinite Geometric Series

70. Slide 9- 70 Example Summing Infinite Geometric Series

71. 9.6 Mathematical Induction

72. Slide 9- 72 Quick Review

73. Slide 9- 73 What you’ll learn about The Tower of Hanoi Problem Principle of Mathematical Induction Induction and Deduction … and why The principle of mathematical induction is a valuable technique for proving combinatorial formulas.

74. Slide 9- 74 The Tower of Hanoi Solution The minimum number of moves required to move a stack of n washers in a Tower of Hanoi game is 2 n – 1.

75. Slide 9- 75 Principle of Mathematical Induction Let P n be a statement about the integer n. Then P n is true for all positive integers n provided the following conditions are satisfied: 1. (the anchor) P 1 is true; 2. (inductive step) if P k is true, then P k+1 is true.

76. Slide 9- 76 Math Induction

77. Slide 9- 77 Math Induction Consider the sequence {a n } defined recursively by a 1 = 2 and a n = a n-1 + 3. Find an explicit formula for a n and prove.

78. Slide 9- 78 Math Induction 1.Prove for n =1 3(1) – 1 = 2 2.Assume for n = k a k = 3k – 1 3.Prove for n = k + 1 1.a k+1 = 3(k + 1) – 1 2.a k+1 = 3k + 3 – 1 3.a k+1 = 3k - 1 + 3 4.a k+1 = a k + 3

79. Slide 9- 79 Math Induction Consider the sequence {a n } defined recursively by a 1 = 3 and a n = a n-1 * 4. Find an explicit formula for a n and prove.

80. Slide 9- 80 Math Induction 1.Prove for n =1 3*4 1-1 = 3*4 0 = 3*1 = 3 2.Assume for n = k a k = 3*4 k-1 3.Prove for n = k + 1 1.a k+1 = 3*4 (k+1)-1 2.a k+1 = 3*4 k-1+1 3.a k+1 = 3*4 k-1 * 4 4.a k+1 = a k * 4

81. Show that is true for all natural numbers 123 1 2 +++ …+= + n nn n (). Step 1: Show true for n = 1 Step 2: Assume true for some number k, determine whether true for k + 1. Math Induction

82. Step 3: Prove for n = k+1

83. Slide 9- 83 Prove that the sum of the first n odd integers equals n 2 1 = 1 = 1 2 1 + 3 = 4 = 2 2 1 + 3 + 5 = 9 = 3 2 1 + 3 + 5 + 7 = 16 = 4 2 Math Induction

84. Slide 9- 84 Prove that the sum of the first n odd integers equals n 2 Matrh Induction 1. Prove for n = 1 1 = 1 2 = 1 2. Assume for n = k 1 + 3 + 5 + …. + 2k – 1 = k 2 3. Prove for n = k + 1 1 + 3 + 5 + … + 2k – 1 + 2(k + 1) – 1 = (k + 1) 2 k 2 + 2k + 2 – 1 = (k + 1) 2 k 2 + 2k + 1 = (k + 1) 2 (k + 1)(k + 1) = (k + 1) 2

85. 9.7 Statistics and Data (Graphical)

86. Slide 9- 86 Quick Review

87. Slide 9- 87 What you’ll learn about Statistics Displaying Categorical Data Stemplots Frequency Tables Histograms Time Plots … and why Graphical displays of data are increasingly prevalent in professional and popular media. We all need to understand them.

88. Slide 9- 88 Leading Causes of Death in the United States in 2001 Cause of DeathNumber of DeathsPercentage Heart Disease700,14229.0 Cancer553,76822.9 Stroke163,5386.8 Other1,018,97741.3 Source: National Center for Health Statistics, as reported in The World Almanac and Book of Facts 2005.

89. Slide 9- 89 Bar Chart, Pie Chart, Circle Graph

90. Slide 9- 90 Example Making a Stemplot Make a stemplot for the given data. 12.3 23.4 12.0 24.5 23.7 18.7 22.4 19.5 24.5 24.6

91. Slide 9- 91 Example Making a Stemplot Make a stemplot for the given data. 12.3 23.4 12.0 24.5 23.7 18.7 22.4 19.5 24.5 24.6 StemLeaf 120,3 187 195 224 234,7 245,5,6

92. Slide 9- 92 Time Plot

93. 9.8 Statistics and Data (Algebraic)

94. Slide 9- 94 Quick Review Solutions

95. Slide 9- 95 What you’ll learn about Parameters and Statistics Mean, Median, and Mode The Five-Number Summary Boxplots Variance and Standard Deviation Normal Distributions … and why The language of statistics is becoming more commonplace in our everyday world.

96. Slide 9- 96 Mean

97. Slide 9- 97 Median The median of a list of n numbers {x 1,x 2,…,x n } arranged in order (either ascending or descending) is the middle number if n is odd, and the mean of the two middle numbers if n is even.

98. Slide 9- 98 Mode The mode of a list of numbers is the number that appears most frequently in the list.

99. Slide 9- 99 Example Finding Mean, Median, and Mode Find the (a) mean, (b) median, and (c) mode of the data: 3, 6, 5, 7, 8, 10, 6, 2, 4, 6

100. Slide 9- 100 Weighted Mean

101. Slide 9- 101 Five-Number Summary

102. Slide 9- 102 Boxplot

103. Slide 9- 103 Outlier A number in a data set can be considered an outlier if it is more than 1.5×IQR below the first quartile or above the third quartile.

104. Slide 9- 104 Data Set L1 Range = 9 Mean = 5.5 Median = 5.5 Data Set L2 Range = 9 Mean = 5.5 Median 5.5 Box and Whisker Plots Find the mean, median, and range for the following data and a box and whisker plot.

105. Slide 9- 105 a measure of variation of the scores about the mean (average deviation from the mean) Standard Deviation

106. Slide 9- 106 calculators can compute the population standard deviation of data 2  ( x - µ ) N  =

107. Slide 9- 107 x 1 2 9 10 x -  -4.5 -3.5 3.5 4.5 (x -  ) 2 20.25 12.25 20.25 Example: Find the Standard Deviation

108. Slide 9- 108 x 1 5 6 10 x -  -4. 5 -. 5. 5 4. 5 (x -  ) 2 20.25. 25 20.25 Example: Find the Standard Deviation

109. Slide 9- 109 Normal Curve

110. Slide 9- 110 The 68-95-99.7 Rule If the data for a population are normally distributed with mean μ and standard deviation σ, then Approximately 68% of the data lie between μ - 1σ and μ + 1σ. Approximately 95% of the data lie between μ - 2σ and μ + 2σ. Approximately 99.7% of the data lie between μ - 3σ and μ + 3σ.

111. Slide 9- 111 The 68-95-99.7 Rule

112. Slide 9- 112 Scatter Plots Scatter Plots: A plot of all the ordered pairs of two variable data on a coordinate axis. Correlation Coefficient (r): The measure of strength between two variables -1 < r < 1

113. Slide 9- 113 Scatter Plots Least-squares Line The least squares line or the line of best fit for a set of n data points is the line described as follows:

114. Slide 9- 114 Scatter Plots Types of correlation Linear y = ax + b Logarithmic y = a + b lnx All x > 0 Exponential y = ab x All y > 0 Power y = ax b All x,y > 0

115. Slide 9- 115 Scatter Plots Find the equation of best fit and use the equation to predict the distance at 9 seconds.

116. Slide 9- 116 Scatter Plots

117. Slide 9- 117 Scatter Plots Conclusion: The power regression model is the best fit d = t 2 At 9 seconds, the distance d = 9 2 = 81 m

118. Slide 9- 118 Chapter Test

119. Slide 9- 119 Chapter Test

120. Slide 9- 120 Chapter Test

121. Slide 9- 121 Chapter Test Solutions

122. Slide 9- 122 Chapter Test Solutions

123. Slide 9- 123 Chapter Test Solutions