Advanced Algebra Chapter 12

1. Advanced Algebra Chapter 12 Probability and Statistics

2. The Fundamental Counting Principle and Permutations—12.1

3. Fundamental Counting Principle • If 2 events: • If one event can occur in m ways and another event can occur in n ways, then the number of ways both events can occur is m * n

4. Fundamental Counting Principle • At Subway, there are 5 bread options and 8 different meat options…. • How many different subs can you get?

5. Fundamental Counting Principle • What about subs with cheeses? • Lets say there are 5 kinds of cheeses, now how many subs? • What about dressings? • Lets say there are 7 different kinds…

6. Fundamental Counting Principle • What about how many subs when veggies are included? • We’ll come back to this…

7. Examples • In Minnesota, license plates have 3 numbers and 3 letters. Excluding Vanity plates, how many combinations are there? • What if repeat digits and letters are not allowed?

8. Factorials • Product of every integer up to the given integer • Examples: • 6! • 4! • 10! • 0!

9. Permutations • An ordering of n amount of objects • When is this important? • Tournament placing • Getting dressed • Etc.

10. Permutations • The number of permutations of r objects taken from a group of n distance objects is denoted • The formula is:

11. Permutations • 12 teams compete for 3 places. How many ways can they place? • How many ways can they finish?

12. Permutations • How many ways can 16 wrestlers place at the state tournament? • How many ways can they finish?

13. Permutations…with repetition • The number if distinguishable permutations of n objects where one object is repeated times, another times, and so on is:

14. Permutations…with repetition • Find the number of distinguishable permutations of the letters in the word: SUMMER

15. Permutations…with repetition • Find the number of distinguishable permutations of the letters in the word: MISSISSIPPI

16. p.705#15-30 All, 31-61 Odd (10 pt)

17. Combinations and the Binomial Theorem—12.2

18. Combinations • A grouping of r objects from a group of n objects where order is not important • Dealing with the same data, will there be more permutations or combinations of that data?

19. Combinations • The number of combinations of r objects taken from a group of n distinct objects is noted by and given by:

20. Combinations • Using a standard deck of cards… • How many ways can you get a flush?

21. Combinations • Using a standard deck of cards… • How many ways can you get 3 of a kind?

22. Combinations • Using a standard deck of cards… • How many 5 cards combinations are possible?

23. Combinations • Using a standard deck of cards… • What is the probability of getting a royal flush?

24. Combinations • A restaurant offers 6 salad toppings. On a deluxe salad, you can have up to 4 toppings. How many combinations are available?

25. Subway Once Again… • 5 Breads, 8 meats, 5 cheeses, 7 dressings…

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27. An Introduction to Probability—12.3

28. Probability • The probability of an event occurring is a number between 0 and 1 that indicates the likelihood the event will occur

29. Probability • When all outcomes are equally likely, the theoretical probability an event will occur is:

30. Examples • Rolling a dice, what is the probability that: • You roll a 2 • You roll an odd number • You roll anything but a 2 • You roll something greater than 4 • You roll less than 1 • You roll less than 7

31. Experimental Probability • Probability based on an experiment, conducting a survey, or looking at the history of an event.

32. Experimental Probability • Favorite Colors in the class…

33. Geometric Probability • Probabilities found by calculating a ratio of two lengths, areas, or volumes • Dealing with where something lands on an object, etc.

34. Geometric Probability • What is the probability an object randomly dropped will land in the red square?

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36. Probability of Compound Events—12.4

37. Compound Events • Union • When you consider the outcomes of two different events • Intersection • Where these two things overlap • Compound Event: • The Union or intersection of the two events

38. Compound Events • What is the difference between “and” and “or”?

39. Compound Events • If A and B are two events, then the probability of A or B is:

40. Examples • What is the probability of getting: • A face card: • A heart: • A heart or a face card:

41. Examples • What is the probability: • Someone is wearing a blue shirt • Someone is wearing a T-shirt • Someone is wearing a blue or a T-shirt

42. Compound Events • Two events are mutually exclusive iff there is no intersection of the two events • Is this equation different???

43. Finding the Complement • Any probability plus it’s complement is always 1 • Everything can be broken into two categories • Dice: Roll a 6, Don’t roll a 6 • Cards: Draw an Ace, Don’t get an Ace • Colors: Blue, not Blue

44. Finding the Complement • A’ is denoted as the complement of A • The probability of the complement of Event A is:

45. Example • Four houses in a neighborhood all have the same model of garage door opener, with each opener having 4096 possible codes. What is the probability that at least two of the four houses have the same code?

46. p.727#16-46 Ev 3rd

47. Probability of Independent and Dependent Events—12.5

48. Independent Events • 2 or more events where the outcome of the first does NOT effect on the occurrence of the 2nd. • Tossing 2 coins: The outcome of the 1st has no effect on the outcome of the 2nd

49. Probability of Independent Events • If A and B are independent events, then the probability of both A and B is:

50. Probability of Independent Events • A game claims that 1 in 15 people will win. What is the probability that you win twice in a row?