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Chapter 9 Review

a n = a 1 + (n – 1)d. a n = a 1 • r (n – 1). Chapter 9 Review. Probability Review (Sections 9.1 – 9.3). Fundamental Principle of Counting (Multiplication Principle). n! = n(n – 1)(n – 2)(n – 3)… (2)(1). How many ways can you arrange:. (5)(4)(3)(2)(1) = 120. A, B, C, D, E. n!

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Chapter 9 Review

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  1. an = a1 + (n – 1)d an = a1• r (n – 1) Chapter 9 Review

  2. Probability Review (Sections 9.1 – 9.3)

  3. Fundamental Principle of Counting (Multiplication Principle) n! = n(n – 1)(n – 2)(n – 3)… (2)(1) How many ways can you arrange: (5)(4)(3)(2)(1) = 120 A, B, C, D, E

  4. n! (n – r)! nPr = 6! (6 – 3)! 720 6 Permutations Order is important!!! using “n” objects to fill “r” blanks in order. permutations of “n” objects taken “r” at a time: How many ways can 6 runners finish 1 – 2 – 3? 6P3 = = = 120

  5. n! (n – r)! • r! nCr = 10! (8)! • 2! 3,628,800 40320 • 2 = Combinations Order is NOT important!!! only interested in the ways to select the “r” objects regardless of the order in which we arrange them. How many ways can 2 cards be picked from a deck of 10: 10C2 = = 45

  6. Subsets of an “n” – set 2n There are ________ subsets of a set with n objects (including the empty set and the entire set). Example: DiMaggio’s Pizzeria offers patrons any combination of up to 10 different pizza toppings. How many different pizzas can be ordered if we can choose any number of toppings (0 through 10)? We could add up all the numbers of the form for r = 0, 1, …, 10 but there is an easier way. In considering each option of a topping, we have 2 choices: __________ or __________. Yes No Therefore the number of different possible pizzas is: 2n = 210 = 1024

  7. Binomial Theorem Don’t forget:

  8. Also, recall that in mathematics, the word or signifies addition; the word and signifies multiplication . Find the probability of selecting an ace or a king from a draw of one card from a standard deck of cards. Find the probability of selecting an ace and a king from a draw of one card from a standard deck of cards.

  9. Venn Diagrams sample space (all students) subsets to represent “girls” and “sports” girls sports “boys” “no sports” students decimals 1 0.36 0.18 0.23 0.23

  10. P(A B) 0.5 1 0.25 0.125 + 0.125 + 0.5 = 0.75 conditional probability dependent the probability of the event A, given that event B occurs 2/4 or 0.5 1 along the branches that come out of the two jars

  11. P(A) • P(B A) the ends of the branches conditional probability formula

  12. binomial distribution binomial Theorem 4C2 = 6 6

  13. Homework Answers # 54, 56, 61, 65, 68, 70, 76, 77 , 17, 23 (p. 748 – 749)

  14. Probability Review Questions

  15. Probability Exercise Suppose there is a 70% chance of rain tomorrow. If it rains, there is a 10% chance that all of the rides at an amusement park will be operating. If it doesn’t rain, there is a 95% chance all of the rides will be operating. What is the probability that all of the rides will be operating tomorrow?

  16. Probability Exercise Binomial Theorem: Bubba rolls a fair die 6 times. What is the probability that he will roll exactly two 2’s?

  17. Probability Exercise There are 20 runners on a track team. How many groups of 4 can be selected to run the 4 x 100 relay? How many ways can 4 runners be selected to run 1st – 2nd – 3rd – 4th?

  18. Probability Exercise A fair coin is tossed 10 times. Find the probability of tossing HHHHHTTTTT. Find the probability of tossing exactly 5 tails in those 10 tosses

  19. Review Question # 43 (p. 748)

  20. Probability Exercise Find the x4 term in the expansion of:

  21. Probability Exercise License plates are created using 3 letters of the alphabet for the first 3 characters and 4 numbers for the last 4 characters. How many possible different license plates are there if the letters and numbers are NOT allowed to repeat?

  22. Probability Exercise • A spinner, numbered 1 through 10, is spun twice. • What is the probability of spinning a 1 and a 10 in any order? • What is the probability of not spinning the same number twice?

  23. Probability Exercise Expand:

  24. Review Question 12, 9.5, 7, 4.5, … 10, 12, 14.4, 17.28, … Is the series arithmetic or geometric? Find the explicit formula Find the recursive formula Find the 100th term Find the sum for a1 through a100 an = 12 – 2.5(n – 1) an = an-1 – 2.5 12 – 2.5(100 – 1) = – 235.5

  25. Review Question 12, 9.5, 7, 4.5, … 10, 12, 14.4, 17.28, … Is the series arithmetic or geometric? Find the explicit formula Find the recursive formula Find the 100th term Find the sum for a1 through a100 an = 10(1.2)(n – 1) an = an-1(2.5) 10(1.2)99 = 690,149,787.7

  26. Review Question Evaluate: – 43 – ½(3)2 – ½(4)2 – ½(5)2 – ½(6)2 – ½(9) – ½(16) – ½(25) – ½(36)

  27. Review Question Is this sequence arithmetic or geometric? 9, 18, … 144

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