1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 1 Homework, Page 715 Expand the binomial using a calculator to find the binomial coefficients. 1.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 2 Homework, Page 715 Expand the binomial using Pascal’s triangle to find the binomial coefficients. 5.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 3 Homework, Page 715 Evaluate the expression by hand before checking your work. 9.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 4 Homework, Page 715 Find the coefficient of the given term in the binomial expansion. 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 5 Homework, Page 715 Use the Binomial Theorem to find the polynomial expansion of the function. 17.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 6 Homework, Page 715 Use the Binomial Theorem to expand each expression. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 7 Homework, Page 715 Use the Binomial Theorem to expand each expression. 25.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 8 Homework, Page 715 29.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 9 Homework, Page 715 33.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 10 Homework, Page 715 37.What is the coefficient of x 4 in the expansion of (2x + 1) 8 ? A.16 B.256 C.1120 D.1680 E.26,680

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.3 Probability

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 12 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.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 13 Probability of an Event (Equally Likely Outcomes)

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 14 Probability Distribution for the Sum of Two Fair Dice OutcomeProbability 21/36 32/36 43/36 54/36 65/36 76/36 85/36 94/36 103/36 112/36 121/36

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 15 Example Rolling the Dice Find the probability of rolling a sum evenly divisible by 4 on a single roll of two fair dice.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 16 Probability Function

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 17 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)?

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 18 Probability of an Event (Outcomes not Equally Likely)

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 19 Example Rolling the Dice Find the probability of rolling a sum evenly divisible by 3 on a single roll of two fair dice.

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 20 Strategy for Determining Probabilities

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 21 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?

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 22 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.

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 23 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.

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 24 Conditional Probability Formula

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 25 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?

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 26 Binomial Distribution

27. 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 all 15?

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 28 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?

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 29 Homework Homework Assignment #32 Read Section 9.4 Page 728, Exercises: 1 - 53 (EOO)

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.4 Sequences

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 31 Quick Review

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 32 Quick Review Solutions

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 33 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.

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 34 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

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 35 Example of an Explicitly Defined Sequence

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 36 Example of a Recursively Defined Sequence

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 37 Limit of a Sequence

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 38 Example Finding Limits of Sequences

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 39 Arithmetic Sequence

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 40 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, …

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 41 Geometric Sequence

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 42 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,…

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 43 Sequences and Graphing Calculators One way to graph an explicitly defined sequence is as a scatter plot of the points of the form (k,a k ). A second way is to use the sequence mode on a graphing calculator.

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 44 Example Graphing a Sequential Scatter Plot Use you calculator to generate the first 10 terms of the sequence explicitly defined by a n = 3n - 5 in a scatter plot.

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 45 Example Calculating Sequence Values Use you calculator to generate the first 10 terms of the sequence explicitly defined by a 1 = 4, a n = 3a n-1 + 5 in a scatter plot.

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 46 The Fibonacci Sequence