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Homework, Page 715 Expand the binomial using a calculator to find the binomial coefficients. 4 1. a b 4 4 0 4 3 1 4 2 2 4 3 4 0 4 a b 0 a b 1 a b 2 a b 3 ab 4 a b 4 a 4 4a 3b1 6a 2b 2 4ab3 b 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 1 Homework, Page 715 Expand the binomial using Pascal’s triangle to find the binomial coefficients. 3 5. x y 1 1 1 1 x y 3 1 2 3 1 3 1 x3 3x 2 y 3xy 2 y 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 2 Homework, Page 715 Evaluate the expression by hand before checking your work. 9. 9 2 9 8 7! 9! 9 2 2! 9 2 ! 2 1 7! 9 4 36 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 3 Homework, Page 715 Find the coefficient of the given term in the binomial expansion. 14 11 3 x y term in x y 13. 14 3 14 C3 364 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 4 Homework, Page 715 Use the Binomial Theorem to find the polynomial expansion of the function. 5 f x x 2 17. f x x 2 5 5 5 5 4 5 3 5 2 2 3 x x 2 x 2 x 2 0 1 2 3 5 5 4 5 x 2 2 4 5 x5 5 x 4 2 10 x3 4 10 x 2 8 5 x 16 32 x5 10 x 4 40 x3 80 x 2 80 x 32 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 5 Homework, Page 715 Use the Binomial Theorem to expand each expression. 4 21. 2x y 4 4 4 4 4 3 2 2 3 4 4 2x y 0 2 x 1 2 x y 2 2 x y 3 2 x y 4 y 16 x 4 4 8 x3 y 6 4 x 2 y 2 4 2 x y 3 y 4 4 16 x 4 32 x3 y 24 x 2 y 2 8 xy 3 y 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 6 Homework, Page 715 Use the Binomial Theorem to expand each expression. 5 2 x 3 25. x 3 5 x 2 5 5 x 2 4 3 5 x 2 3 32 5 x 2 2 33 0 1 2 3 5 2 5 5 4 x 3 3 4 5 x 10 5 x 8 3 10 x 6 9 10 x 4 27 5 x 2 81 243 2 5 x 10 15 x 8 90 x 6 270 x 4 405 x 2 243 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 7 Homework, Page 715 n n n for all integers n 1. 29. Prove that 1 n 1 n! n n n 1! n 1 1! n 1! 1 n 1! n n 1! n! n n 1 n 1! n n 1 ! n 1!1! n n n 1 n 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley n Slide 9- 8 Homework, Page 715 n n 1 2 Prove that n for all integers n 2. 33. 2 2 n n 1 n 2 ! n n 1 n 2 n n! n 2 2! n 2 ! 2 2 n 2 ! 2 n 1! n 1! n 1 n n 1! n 2 n n 1 2 2! n 1 2 ! 2 n 1! 2 2 n 1! n 2 n n 2 n n 2 n n 2 n 2n 2 n2 2 2 2 2 n n 1 2 n 2 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 9 Homework, Page 715 37. A. B. C. D. E. What is the coefficient of x4 in the expansion of (2x + 1)8? 16 256 8 4 4 4 1120 70 16x 2 x 1 1120x 4 4 1680 26,680 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 10 9.3 Probability Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 12 Probability of an Event (Equally Likely Outcomes) If E is an event in a finite, nonempty sample space S of equally likely outcomes, then the probability of the event E is the number of outcomes in E P( E ) . the number of outcomes in S Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 13 Probability Distribution for the Sum of Two Fair Dice Outcome 2 3 4 5 6 7 8 9 10 11 12 Probability 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 1 2 3 4 5 6 1 2 3 4 5 6 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 3 4 5 6 7 8 3 4 5 6 4 5 6 7 5 6 7 8 6 7 8 9 7 8 9 10 8 9 10 11 9 10 11 12 Slide 9- 14 Example Rolling the Dice Find the probability of rolling a sum evenly divisible by 4 on a single roll of two fair dice. 1 2 3 4 5 6 1 2 3 4 5 6 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 3 4 5 6 7 8 3 4 5 6 4 5 6 7 5 6 7 8 6 7 8 9 7 8 9 10 8 9 10 11 9 10 11 12 Slide 9- 15 Probability Function A probability function is a function P that assigns a real number to each outcome in a sample space S subject to the following conditions: 1. 0 P(O) 1; 2. the sum of the probabilities of all outcomes in S is 1; 3. P() 0. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 16 Example Testing the Validity of a Probability Function Is it possible to weight a standard number cube in such a way that the probability of rolling a number n is exactly 1/(n+2)? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 17 Probability of an Event (Outcomes not Equally Likely) Let S be a finite, nonempty sample space in which every outcome has a probability assigned to it by a probability function P. If E is any event in S , the probability of the event E is the sum of the probabilities of all the outcomes contained in E. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 18 Example Rolling the Dice Find the probability of rolling a sum evenly divisible by 3 on a single roll of two fair dice. 1 2 3 4 5 6 1 2 3 4 5 6 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 3 4 5 6 7 8 3 4 5 6 4 5 6 7 5 6 7 8 6 7 8 9 7 8 9 10 8 9 10 11 9 10 11 12 Slide 9- 19 Strategy for Determining Probabilities 1. Determine the sample space of all possible outcomes. When possible, choose outcomes that are equally likely. 2. If the sample space has equally likely outcomes, the probability of an the number of outcomes in E event E is determined by counting: P ( E ) . the number of outcomes in S 3. If the sample space does not have equally likely outcomes, determine the probability function. (This is not always easy to do.) Check to be sure that the conditions of a probability function are satisfied. Then the probability of an event E is determined by adding up the probabilities of all the outcomes contained in E. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 20 Example Choosing Chocolates Dylan opens a box of a dozen chocolate crèmes and offers three of them to Russell. Russell likes vanilla crèmes the best, but all the chocolates look alike on the outside. If five of the twelve crèmes are vanilla, what is the probability that all of Russell’s picks are vanilla? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 21 Multiplication Principle of Probability Suppose an event A has probability p1 and an event B has probability p2 under the assumption that A occurs. Then the probability that both A and B occur is p1p2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 22 Venn Diagram Venn diagrams are visual representations of groupings of events. For example, if 63% of the students are girls and 54% of the students play sports, find the percentage of boys playing sports if 1/3 of the girls play sports. Girls Sports Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 23 Conditional Probability Formula If the event B depends on the event A, then P( B | A) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P( A and B) . P( A) Slide 9- 24 Example Using the Conditional Probability Formula Two identical cookie jars are on a counter. Jar A contains eight cookies, six of which are oatmeal, and jar B contains four cookies, two of which are oatmeal. If an oatmeal cookie is selected, what is the likelihood it came from the jar A? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 25 Binomial Distribution Suppose an experiment consists of n-independent repetitions of an experiment with two outcomes, called "success" and "failure." Let P (success) p and P (failure) q. (Note that q 1 p.) Then the terms in the binomial expansion of ( p q ) n give the respective probabilities of exactly n, n 1,..., 2, 1, 0 successes. Number of successes out of Probability n independent repetitions n pn n 1 n n1 n 1 p q 1 0 n n1 r pq qn Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 26 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? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 27 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? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 28 Homework Homework Assignment #32 Read Section 9.4 Page 728, Exercises: 1 - 53 (EOO) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 29 9.4 Sequences Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review Evaluate each expression when a 3, r 2, n 4 and d 2. 1. a (n 1)d 2. a r Find a . n 1 10 k 1 k 4. a 2 3 3. a k k 1 k 5. a a 3 and a 10 k k 1 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 31 Quick Review Solutions Evaluate each expression when a 3, r 2, n 4 and d 2. 1. a (n 1)d 9 2. a r 24 Find a . n 1 10 k 1 k 4. a 2 3 3. a k k 1 k 11 10 39,366 5. a a 3 and a 10 k k 1 9 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 32 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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 33 Sequence Sequence - an ordered progression of numbers Finite sequence - a sequence with a finite number of entries Infinite sequence - a sequence that continues without bound Explicitly defined sequence - a sequence for which any entry may be written directly using the definition Recursively defined sequence - a sequence defined in such a manner that one must know the prior entry before being able to write the next entry using the definition Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 34 Example of an Explicitly Defined Sequence Find the first 4 and the 50 th term of the sequence ak in which ak k 2 3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 35 Example of a Recursively Defined Sequence Find the first 4 and the 50th term of the sequence ak in which a1 2, an an 1 2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 36 Limit of a Sequence Let an be a sequence of real numbers, and consider lim an . n If the limit is a finite number L, the sequence converges and L is the limit of the sequence. If the limit is infinite or nonexistent, the sequence diverges. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 37 Example Finding Limits of Sequences Determine whether the sequence converges or diverges. If it converges, give the limit. 2 1 2 2 2,1, , , ,..., ,... 3 2 5 n Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 38 Arithmetic Sequence A sequence an is an arithmetic sequence if it can be written in the form a, a d , a 2d ,..., a (n 1)d ,... for some constant d . The number d is called the common difference. Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d : an an1 d (for all n 2). Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 39 Example Arithmetic Sequences Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term. -2, 1, 4, 7, … Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 40 Geometric Sequence A sequence an is a geometric sequence if it can be written in the form a, a r , a r 2 ,..., a r n1 ,... for some nonzero constant r. The number r is called the common ratio. Each term in a geometric sequence can be obtained recursively from its preceding term by multiplying by r : an an1 r (for all n 2). Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 41 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,… Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 42 Sequences and Graphing Calculators One way to graph an explicitly defined sequence is as a scatter plot of the points of the form (k,ak). A second way is to use the sequence mode on a graphing calculator. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 43 Example Graphing a Sequential Scatter Plot Use you calculator to generate the first 10 terms of the sequence explicitly defined by an = 3n - 5 in a scatter plot. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 44 Example Calculating Sequence Values Use you calculator to generate the first 10 terms of the sequence explicitly defined by a1 = 4, an = 3an-1 + 5 in a scatter plot. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 45 The Fibonacci Sequence The Fibonacci sequences can be defined recursively by a1 1 a2 1 an an2 an1 for all positive integers n 3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 46