1. 1/31/2007 Pre-Calculus Chapter 9 Review a n = a 1 + (n – 1)d a n = a 1 r (n – 1)

2. 1/31/2007 Pre-Calculus Probability Review (Sections 9.1 – 9.3) Probability Review (Sections 9.1 – 9.3)

3. 1/31/2007 Pre-Calculus Fundamental Principle of Counting (Multiplication Principle) Fundamental Principle of Counting (Multiplication Principle) n! = n(n – 1)(n – 2)(n – 3)… (2)(1) A, B, C, D, E (5)(4)(3)(2)(1) = 120 How many ways can you arrange:

4. 1/31/2007 Pre-Calculus Permutations Order is important!!! Permutations Order is important!!! permutations of “n” objects taken “r” at a time: using “n” objects to fill “r” blanks in order. How many ways can 6 runners finish 1 – 2 – 3? n P r = n! (n – r)! 6 P 3 = 6! (6 – 3)! 720 6 120 = =

5. 1/31/2007 Pre-Calculus 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: n C r = n! (n – r)! r! 10 C 2 = = 45 10! (8)! 2! = 3,628,800 40320 2 Combinations Order is NOT important!!! Combinations Order is NOT important!!!

6. 1/31/2007 Pre-Calculus Subsets of an “n” – set There are ________ subsets of a set with n objects (including the empty set and the entire set). 2n2n 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 __________. Therefore the number of different possible pizzas is: Example: Yes No 2 n = 2 10 = 1024

7. 1/31/2007 Pre-Calculus Binomial Theorem Don’t forget:

8. 1/31/2007 Pre-Calculus 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. 1/31/2007 Pre-Calculus Venn Diagrams sample space (all students) subsets to represent “girls” and “sports” “boys” “no sports” decimals 1 students girls sports 0.360.180.23

10. 1/31/2007 Pre-Calculus 0.5 conditional probability 0.25 1 0.125 + 0.125 + 0.5 = 0.75 dependent 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 the probability P(A B)

11. 1/31/2007 Pre-Calculus the ends of the branches conditional probability formula P(A) P(B A)

12. 1/31/2007 Pre-Calculus binomial distribution binomial Theorem 4 C 2 = 6 6

13. 1/31/2007 Pre-Calculus Homework Answers # 54, 56, 61, 65, 68, 70, 76, 77, 17, 23 (p. 748 – 749)

14. 1/31/2007 Pre-Calculus Probability Review Questions Probability Review Questions

15. 1/31/2007 Pre-Calculus 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. 1/31/2007 Pre-Calculus Probability Exercise Binomial Theorem: Bubba rolls a fair die 6 times. What is the probability that he will roll exactly two 2’s?

17. 1/31/2007 Pre-Calculus 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 1 st – 2 nd – 3 rd – 4 th ?

18. 1/31/2007 Pre-Calculus 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. 1/31/2007 Pre-Calculus Review Question # 43 (p. 748)

20. 1/31/2007 Pre-Calculus Probability Exercise Find the x 4 term in the expansion of:

21. 1/31/2007 Pre-Calculus 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. 1/31/2007 Pre-Calculus 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. 1/31/2007 Pre-Calculus Probability Exercise Expand:

24. 1/31/2007 Pre-Calculus a n = 12 – 2.5(n – 1) a n = a n-1 – 2.5 Review Question Is the series arithmetic or geometric? Find the explicit formula Find the recursive formula Find the 100 th term Find the sum for a 1 through a 100 12, 9.5, 7, 4.5, … 10, 12, 14.4, 17.28, … 12 – 2.5(100 – 1) = – 235.5

25. 1/31/2007 Pre-Calculus a n = 10(1.2) (n – 1) Review Question Is the series arithmetic or geometric? Find the explicit formula Find the recursive formula Find the 100 th term Find the sum for a 1 through a 100 12, 9.5, 7, 4.5, … 10, 12, 14.4, 17.28, … a n = a n-1 ( 2.5) 10(1.2) 99 = 690,149,787.7

26. 1/31/2007 Pre-Calculus Review Question Evaluate: – ½(3) 2 – ½(4) 2 – ½(5) 2 – ½(6) 2 – ½(9) – ½(16) – ½(25) – ½(36) – 43

27. 1/31/2007 Pre-Calculus Review Question Is this sequence arithmetic or geometric? 9, 18, … 144