Slide 9- 1

Chapter 9 Discrete Mathematics

9.1 Basic Combinatorics

Slide 9- 4 Quick Review

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.

Slide 9- 6 Multiplication Principle of Counting

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.

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

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.

Slide Distinguishable Permutations

Slide Permutations Counting Formula

Slide Combination Counting Formula

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

Slide Formula for Counting Subsets of an n-Set

9.2 The Binomial Theorem

Slide Quick Review

Slide 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.

Slide Binomial Coefficient

Slide Example Using n C r to Expand a Binomial

Slide Recursion Formula for Pascal’s Triangle

Slide The Binomial Theorem

Slide Basic Factorial Identities

9.3 Probability

Slide Quick Review Solutions

Slide 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.

Slide Probability of an Event (Equally Likely Outcomes)

Slide (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

Slide 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 Find the probability that the sum is a 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

Slide 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

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

Slide Probability Function

Slide Probability of an Event (Outcomes not Equally Likely)

Slide Strategy for Determining Probabilities

Slide 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?

Slide 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.

Slide 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?

Slide 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?

Slide Conditional Probability Formula

Slide 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

Slide 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

Slide  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

Slide 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.

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

Slide 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) = b) P(at least 3) = P(3 or 4 or 5) = P(3) or P(4) or P(5) = = Binomial Theorem

Slide 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:

Slide 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?

Slide 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?

Slide 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?

Slide 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?

9.4 Sequences

Slide Quick Review

Slide 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.

Slide Limit of a Sequence

Slide Example Finding Limits of Sequences

Slide Arithmetic Sequence

Slide 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, …

Slide Geometric Sequence

Slide 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,…

Slide 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.

Slide The Fibonacci Sequence

9.5 Series

Slide Quick Review

Slide 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.

Slide Summation Notation

Slide Sum of a Finite Arithmetic Sequence

Slide Example Summing the Terms of an Arithmetic Sequence

Slide Sum of a Finite Geometric Sequence

Slide Infinite Series

Slide Sum of an Infinite Geometric Series

Slide Example Summing Infinite Geometric Series

9.6 Mathematical Induction

Slide Quick Review

Slide 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.

Slide 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.

Slide 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.

Slide Math Induction

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

Slide 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 a k+1 = 3(k + 1) – 1 2.a k+1 = 3k + 3 – 1 3.a k+1 = 3k a k+1 = a k + 3

Slide 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.

Slide 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 a k+1 = 3*4 (k+1)-1 2.a k+1 = 3*4 k a k+1 = 3*4 k-1 * 4 4.a k+1 = a k * 4

Show that is true for all natural numbers …+= + 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

Step 3: Prove for n = k+1

Slide Prove that the sum of the first n odd integers equals n 2 1 = 1 = = 4 = = 9 = = 16 = 4 2 Math Induction

Slide 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 …. + 2k – 1 = k 2 3. Prove for n = k … + 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

9.7 Statistics and Data (Graphical)

Slide Quick Review

Slide 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.

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

Slide Bar Chart, Pie Chart, Circle Graph

Slide Example Making a Stemplot Make a stemplot for the given data

Slide Example Making a Stemplot Make a stemplot for the given data StemLeaf 120, ,7 245,5,6

Slide Time Plot

9.8 Statistics and Data (Algebraic)

Slide Quick Review Solutions

Slide 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.

Slide Mean

Slide 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.

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

Slide 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

Slide Weighted Mean

Slide Five-Number Summary

Slide Boxplot

Slide 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.

Slide 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.

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

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

Slide x x -  (x -  ) Example: Find the Standard Deviation

Slide x x -  (x -  ) Example: Find the Standard Deviation

Slide Normal Curve

Slide The 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.

Slide The Rule

Slide 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

Slide 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:

Slide 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

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

Slide Scatter Plots

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

Slide Chapter Test

Slide Chapter Test

Slide Chapter Test

Slide Chapter Test Solutions

Slide Chapter Test Solutions

Slide Chapter Test Solutions