1. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Five Elementary Probability Theory

2. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Probability Probability is a numerical measurement of likelihood of an event. The probability of any event is a number between zero and one. Events with probability close to one are more likely to occur. If an event has probability equal to one, the event is certain to occur.

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 Probability Notation If A represents an event, P(A) represents the probability of A.

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 Three methods to find probabilities: Intuition (subjective) Relative frequency (experimental) Equally likely outcomes (theoretical)

5. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 Intuition method (subjective; based on opinion or experience) based upon our level of confidence in the result Example: I am 95% sure that I will attend the party.

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 Probability as Relative Frequency (experimental; what has just happened) Probability of an event = the fraction of the time that the event occurred in the past = f n where f = frequency of an event n = sample size

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 Example of Probability as Relative Frequency If you note that 57 of the last 100 applicants for a job have been female, the probability that the next applicant is female would be:

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 Law of Large Numbers In the long run, as the sample size increases and increases, the relative frequencies of outcomes get closer and closer to the theoretical (or equally likely) probability value.

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 Equally likely outcomes (theoretical; if all results have the same chance) No one result is expected to occur more frequently than any other.

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 Three methods to find probabilities: Intuition (subjective) Relative frequency (experimental) Equally likely outcomes (theoretical)

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Probability of an event when outcomes are equally likely =

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Example of Equally Likely Outcome Method When rolling a die, the probability of getting a number less than three =

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Statistical Experiment activity that results in a definite outcome

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Sample Space set of all possible outcomes of an experiment

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Sample Space for the rolling of an ordinary die: 1, 2, 3, 4, 5, 6

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 Three methods to find probabilities: Intuition (subjective) Relative frequency (experimental) Equally likely outcomes (theoretical)

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 For the experiment of rolling an ordinary die: P(even number) = P(result less than four) = P(not getting a two) = 3 = 1 6 2 3 = 1 6 2 5656

18. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Complement of Event A the event not A

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Probability of a Complement P(not A) = 1 – P(A)

20. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 Probability of a Complement If the probability that it will snow today is 30%, P(It will not snow) = 1 – P(snow) = 1 – 0.30 = 0.70

21. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 Slide 14- 21 Formal Probability (cont.) Complement Rule:  The set of outcomes that are not in the event A is called the complement of A, denoted A C.  The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(A C )

22. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 Probabilities of an Event and its Complement Denote the probability of an event by the letter p. Denote the probability of the complement of the event by the letter q. p + q must equal 1 q = 1 - p

23. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 Probability Related to Statistics Probability makes statements about what will occur when samples are drawn from a known population. Statistics describes how samples are to be obtained and how inferences are to be made about unknown populations.

24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 Slide 14- 24 Probability The probability of an event is its long-run relative frequency. –While we may not be able to predict a particular individual outcome, we can talk about what will happen in the long run. For any random phenomenon, each attempt, or trial, generates an outcome. –Something happens on each trial, and we call whatever happens the outcome. –These outcomes are individual possibilities, like the number we see on top when we roll a die.

25. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 Slide 14- 25 Probability (cont.) Sometimes we are interested in a combination of outcomes (e.g., a die is rolled and comes up even). –A combination of outcomes is called an event. When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. –Roughly speaking, this means that the outcome of one trial doesn’t influence or change the outcome of another. –For example, coin flips are independent.

26. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Independent Events The occurrence (or non-occurrence) of one event does not change the probability that the other event will occur.

27. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 If events A and B are independent, P(A and B) = P(A) P(B)

28. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 Slide 14- 28 Formal Probability (cont.) Multiplication Rule: –For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events. –P(A and B) = P(A) x P(B), provided that A and B are independent.

29. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 Slide 14- 29 Formal Probability (cont.) Multiplication Rule: –Two independent events A and B are not disjoint, provided the two events have probabilities greater than zero:

30. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 Conditional Probability If events are dependent, the occurrence of one event changes the probability of the other. The notation P(A|B) is read “the probability of A, given B.” P(A, given B) equals the probability that event A occurs, assuming that B has already occurred.

31. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 For Dependent Events: P(A and B) = P(A) P(B, given A) P(A and B) = P(B) P(A, given B)

32. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 The Multiplication Rules: For independent events: P(A and B) = P(A) P(B) For dependent events: P(A and B) = P(A) P(B, given A) P(A and B) = P(B) P(A, given B)

33. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 For independent events: P(A and B) = P(A) P(B) When choosing two cards from two separate decks of cards, find the probability of getting two fives. P(two fives) = P(5 from first deck and 5 from second) =

34. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 For dependent events: P(A and B) = P(A) P(B, given A) When choosing two cards from a deck without replacement, find the probability of getting two fives. P(two fives) = P(5 on first draw and 5 on second) =

35. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 “And” versus “or” And means both events occur together. Or means that at least one of the events occur.

36. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 For any events A and B, P(A or B) = P(A) + P(B) – P(A and B)

37. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 When choosing a card from an ordinary deck, the probability of getting a five or a red card: P(5 ) + P(red) – P(5 and red) =

38. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 When choosing a card from an ordinary deck, the probability of getting a five or a six: P(5 ) + P(6) – P(5 and 6) =

39. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 For any mutually exclusive events A and B, P(A or B) = P(A) + P(B)

40. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40 When rolling an ordinary die: P(4 or 6) =

41. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 41 Survey results: P(male and college grad) = ?

42. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 42 Survey results: P(male and college grad) =

43. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 43 Survey results: P(male or college grad) = ?

44. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 44 Survey results: P(male or college grad) =

45. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 45 Survey results: P(male, given college grad) = ?

46. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 46 Survey results: P(male, given college grad) =

47. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 47 Can you write three (3) probability questions from the following data? The makers of the movie Titanic imply that lower-class passengers were treated unfairly when the lifeboats were being filled. We want to determine whether that portrayal is accurate. The following table contains the survival data by passenger class for the 1316 passengers. Now trade papers with someone and see if you can answer their questions correctly ; ) ClassSurvivedLost First203122 Second118167 Third178528

48. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 48 Counting Techniques Tree Diagram Multiplication Rule Permutations Combinations

49. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 49 Tree Diagram a method of listing outcomes of an experiment consisting of a series of activities

50. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 50 Tree diagram for the experiment of tossing two coins start H T H H T T

51. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 51 Find the number of paths without constructing the tree diagram: Experiment of rolling two dice, one after the other and observing any of the six possible outcomes each time. Number of paths = 6 x 6 = 36

52. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 52 Multiplication of Choices If there are n possible outcomes for event E 1 and m possible outcomes for event E 2, then there are n x m or nm possible outcomes for the series of events E 1 followed by E 2.

53. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 53 Area Code Example Until a few years ago a three-digit area code was designed as follows. The first could be any digit from 2 through 9. The second digit could be only a 0 or 1. The last could be any digit. How many different such area codes were possible? 8  2  10 = 160

54. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 54 Ordered Arrangements In how many different ways could four items be arranged in order from first to last? 4  3  2  1 = 24

55. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 55 Factorial Notation n! is read "n factorial" n! is applied only when n is a whole number. n! is a product of n with each positive counting number less than n

56. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 56 Calculating Factorials 5! = 5 4 3 2 1 = 3! = 3 2 1 = 120 6

57. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 57 Definitions 1! = 1 0! = 1

58. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 58 Complete the Factorials: 4! = 10! = 6! = 15! = 24 3,628,800 720 1.3077 x 10 12

59. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 59 Permutations A permutation is an arrangement in a particular order of a group of items. There are to be no repetitions of items within a permutation.)

60. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 60 Listing Permutations How many different permutations of the letters a, b, c are possible? Solution: There are six different permutations: abc, acb, bac, bca, cab, cba.

61. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 61 Listing Permutations How many different two-letter permutations of the letters a, b, c, d are possible? Solution: There are twelve different permutations: ab, ac, ad, ba, ca, da, bc, bd, cb, db, cd, dc.

62. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 62 Permutation Formula The number of ways to arrange in order n distinct objects, taking them r at a time, is:

63. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 63 Another notation for permutations:

64. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 64 Find P 7, 3

65. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 65 Applying the Permutation Formula P 3, 3 = _______ P 4, 2 = _______ P 6, 2 = __________ P 8, 3 = _______ P 15, 2 = _______ 612 30336 210

66. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 66 Application of Permutations A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen and arranged in order from #1 to #5? Solution: P 8,5 = = 8 7 6 5 4 = 6720

67. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 67 Combinations A combination is a grouping in no particular order of items.

68. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 68 Combination Formula The number of combinations of n objects taken r at a time is:

69. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 69 Other notations for combinations:

70. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 70 Find C 9, 3

71. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 71 Applying the Combination Formula C 5, 3 = ______ C 7, 3 = ________ C 3, 3 = ______ C 15, 2 = ________ C 6, 2 = ______ 35 1 105 10 15

72. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 72 Application of Combinations A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen if order makes no difference? Solution: C 8,5 = = 56