Review

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(EC or FC) P(EC and FC) P(EC 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) =0.1 P(EC or FC) P(EC and FC) P(EC 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) =0.1 P(EC or FC)=0.9 P(EC and FC) P(EC 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) =0.1 P(EC or FC)=0.9 P(EC and FC)=0.4 P(EC 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) =0.1 P(EC or FC)=0.9 P(EC and FC)=0.4 P(EC and F)= 0.2

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

8. 3.15

9. 3.15

10. 3.15

11. 3.20

12. 3.20

13. 3.20

14. 3.44

15. 3.44

16. 3.44

17. 3.45

18. 3.45

19. 3.45

20. 3.45

21. 3.45

22. 3.45

23. 3.54

24. 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? A B 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. A B 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. A B 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.