1. Review Probabilities –Definitions of experiment, event, simple event, sample space, probabilities, intersection, union compliment –Finding Probabilities –Drawing Venn Diagrams –If A and B are two events then P(A or B) = P(A) + P(B) - P(A and B), P(not A) = 1 - P(A). –Two events A and B are mutually exclusive if P(A and B) = 0.

2. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) P(E C or F C ) P(E C and F C )P(E C and F)

3. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C ) P(E C and F C )P(E C and F)

4. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C )=0.9 P(E C and F C )P(E C and F)

5. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C )=0.9 P(E C and F C )=0.4P(E C and F)

6. Example Suppose P (E) = 0.4, P (F) = 0.3, and P(E or F)=0.6. Find: P(E and F) =0.1P(E C or F C )=0.9 P(E C and F C )=0.4P(E C and F)= 0.2

7. 7 Problems 3.15, 3.20, 3.44, 3.45, 3.54

8. 3.15

11. 3.20

14. 3.44

17. 3.45

18. LowMediumHigh On0.500.100.05 Off0.250.070.03

19. 3.45 LowMediumHighTotal On0.500.100.050.65 Off0.250.070.030.35 Total0.750.170.081

20. 3.45 LowMediumHighTotal On0.500.100.050.65 Off0.250.070.030.35 Total0.750.170.081

21. 3.45 LowMediumHighTotal On0.500.100.050.65 Off0.250.070.030.35 Total0.750.170.081

22. 3.45 LowMediumHighTotal On0.500.100.050.65 Off0.250.070.030.35 Total0.750.170.081

23. 3.54

25. Conditional Probability You are dealt two cards from a deck. What is the probability the first card dealt is a Jack?

26. Conditional Probability You are dealt two cards from a deck. What is the probability the first card dealt is a Jack?

27. Conditional Probability You are dealt two cards from a deck. What is the probability of the second card was a jack given the first card was not a jack is called card dealt is a Jack?

28. Conditional Probability You are dealt two cards from a deck. What is the probability of the second card was a jack given the first card was not a jack is called card dealt is a Jack?

29. Conditional Probability You are dealt two cards from a deck. What is the probability of the second card was a jack given the first card was not a jack is called card dealt is a Jack?

30. Conditional Probability You are dealt two cards from a deck. What is the probability of the second card was a jack given the first card was not a jack is called card dealt is a Jack? The probability of drawing jack given the first card was not a jack is called conditional probability. A key words to look for is “given that.”

31. Conditional Probability The probability of drawing jack given the first card was not a jack is called conditional probability. A key words to look for is “given that.” The probability that the event A occurs, given that B occurs is denoted: This is read the probability of A given B.

32. Conditional Probability You are dealt two cards from a deck. What is the probability the second card dealt is a Jack?

33. Conditional Probability You are dealt two cards from a deck. What is the probability the second card dealt is a Jack?

34. Conditional Probability You are dealt two cards from a deck. What is the probability the second card dealt is a Jack?

35. Example A local union has 8 members, 2 of whom are women. Two are chosen by a lottery to represent the union.

36. Example A local union has 8 members, 2 of whom are women. Two are chosen by a lottery to represent the union. a) What is the probability they are both male?

37. Example A local union has 8 members, 2 of whom are women. Two are chosen by a lottery to represent the union. a) What is the probability they are both male?

38. Example A local union has 8 members, 2 of whom are women. Two are chosen by a lottery to represent the union. a) What is the probability they are both male?

39. Example A local union has 8 members, 2 of whom are women. Two are chosen by a lottery to represent the union. b) What is the probability they are both female?

40. Example A local union has 8 members, 2 of whom are women. Two are chosen by a lottery to represent the union. b) What is the probability they are both female?

41. Conditional Probability How would we draw the event A given B? AB A and B

42. Conditional Probability How would we draw the event A given B? Since we know B has occurred, we ignore everything else. AB A and B

43. Conditional Probability How would we draw the event A given B? Since we know B has occurred, we ignore everything else. AB A and B

44. Conditional Probability How would we draw the event A given B? Since we know B has occurred, we ignore everything else. With some thought this tells us: B A and B

45. Conditional Probability Since we know B has occurred, we ignore everything else. Or rearranging: B A and B

46. Example Find the probability of selecting an all male jury from a group of 30 jurors, 21 of whom are men.

47. Example Find the probability of selecting an all male jury from a group of 30 jurors, 21 of whom are men. Solution: P(12 M) =P(M)*P(M|M)*P(M|MM) * …. = 21/30 * 20/29 * 19/28 * 18/27 * … 10/19 = 0.00340

48. Independent Events Two events A and B are independent if the occurrence of one does not affect the probability of the other.

49. Independent Events Two events A and B are independent if the occurrence of one does not affect the probability of the other. Two events A and B are independent then P(A|B) = P(A).

50. Independent Events Two events A and B are independent if the occurrence of one does not affect the probability of the other. Two events A and B are independent then P(A|B) = P(A). Two events which are not independent are dependent.

51. Multiplication Rule Multiplication Rule: For any pair of events: P(A and B) = P(A) * P(B|A)

52. Multiplication Rule Multiplication Rule: For any pair of events: P(A and B) = P(A) * P(B|A) For any pair of independent events: P(A and B) = P(A) * P(B)

53. Multiplication Rule For any pair of events: P(A and B) = P(A) * P(B|A) For any pair of independent events: P(A and B) = P(A) * P(B) If P(A and B) = P(A) * P(B), then A and B are independent.

54. Multiplication Rule Multiplication Rule: P(A and B) = P(A) * P(B) if A and B are independent. P(A and B) = P(B) * P(A|B) if A and B are dependent. Note: The multiplication rule extends to several events: P(A and B and C) =P(C)*P(B|C)*P(A|BC)

55. Example A study of 24 mice has classified the mice by two categories BlackWhiteGrey Eye Colour Red Eyes352 Black Eyes176 Fur Colour

56. A study of 24 mice has classified the mice by two categories a) What is the probability that a randomly selected mouse has white fur? b) What is the probability it has black eyes given that it has black fur? c) Find pairs of mutually exclusive and independent events. BlackWhiteGrey Eye Colour Red Eyes352 Black Eyes176 Fur Colour

57. A study of 24 mice has classified the mice by two categories a) What is the probability that a randomly selected mouse has white fur?12/24=0.5 b) What is the probability it has black eyes given that it has black fur? c) Find pairs of mutually exclusive and independent events. BlackWhiteGrey Eye Colour Red Eyes352 Black Eyes176 Fur Colour

58. A study of 24 mice has classified the mice by two categories a) What is the probability that a randomly selected mouse has white fur?12/24=0.5 b) What is the probability it has black eyes given that it has black fur?1/4=0.25 c) Find pairs of mutually exclusive and independent events. BlackWhiteGrey Eye Colour Red Eyes352 Black Eyes176 Fur Colour

59. b) What is the probability it has black eyes given that it has black fur?1/4=0.25 c) Find pairs of mutually exclusive and independent events. ME: Black eyes and Red eyes IND: White Fur and Red Eyes; Black Fur and Red Eyes BlackWhiteGrey Eye Colour Red Eyes352 Black Eyes176 Fur Colour

60. Descriptive Phrases Descriptive Phrases require special care! – At most – At least – No more than – No less than

61. Review Conditional Probabilities Independent events Multiplication Rule Tree Diagrams

62. 62 Homework Review Chapter 3 Read Chapter 4.1-4.4 Quiz on Tuesday on Chapters 1 and 2 Problems on next slide

63. Problems Problems 3.66, 3.68, 3.70, 3.75, 3.80, 3.87