Probability Theory II: Addition and Multiplication Rules

1. Applied Probability and Statistics(MATH 301 )Lecture 6: Probability Theory II Addition and Multiplication Rules,Conditional Probability Instructor : Assoc. Prof. Dr. Gamal M. Abdel-Hamid Email : gmabrouk@hotmail.com

2. Recall • A probability experiment is a chance process that leads to well-defined results called outcomes. • E.g.: • Flipping a coin once. • Rolling a die once. • Rolling a die twice. • etc.

3. Recall • The set of all possible outcomes of a probability experiment is called a sample space. • We are going to denote the sample space by S.

4. Recall Examples • Experiment: Tossing a coin once. S = {H, T} • Experiment: Rolling a die once. S = {1, 2, 3, 4, 5, 6} • Experiment: Tossing a coin twice. S = {HH, HT, TH, TT}

5. Recall • An event, E, is a set of outcomes of a probability experiment. • E  S

6. Recall • The classicalprobability, P(E), of an event E is given by

7. Objectives • At the end of this lecture, we will be able to • Determine whether two events are mutually exclusive. • Apply the addition rules. • Determine whether two events are independent. • Apply the multiplication rule for independent events. • Apply the rule of multiplication of dependent events. • Apply the multiplication rule of conditional probability. • Apply the total probability rule.

8. Mutually Exclusive Events • Two events are mutually exclusive if no outcome of the probability experiment would count as an occurrence of both events.

9. Example When drawing a card: • Drawing a club and drawing a heart are mutually exclusive events. • Drawing a club and drawing a king are not mutually exclusive • Drawing the king of clubs is an occurrence of both events.

10. More precisely

11. Example • Rolling a single die. • E1 is the event of getting a prime number. • E2 is the event of getting an even number. • E1 = {2, 3, 5} • E2 = {2, 4, 6}

12. Example • Rolling a single die. • E1 is the event of getting a prime number. • E2 is the event of getting an even number. • E1 = {2, 3, 5} • E2 = {2, 4, 6} • E1 and E2 are not mutually exclusive.

13. Addition Rules • If E1 and E2 are mutually exclusive events, then the probability that E1 or E2 will occur is

14. Mutually Exclusive EventsP(A Or B) = P(A)+P(B)

15. Addition Rule • If E1 and E2 are not mutually exclusive then

16. Not Mutually Exclusive Events • P(A Or B) = P(A)+P(B) – P(A And B) • Otherwise the intersection area would be counted twice

17. Note • The event of E1 or E2’s occurring is E1  E2 • The event of E1 and E2’s occurring is E1  E2

19. Example • A single die is rolled. What is the probability of getting a prime number or an even number?

20. Solution • E1 = {2, 3, 5}, is the event of getting a prime number. P(E1) = 3 / 6 = 1 /2 • E2 = {2, 4, 6} is the event of getting an even number. P(E2) = 3 / 6 = 1 / 2 • Not mutually exclusive. • E1 E2 = {2} P(E1  E2) = 1 / 6 • P(E1  E2) = 1/2 + 1/2  1/6 = 5 / 6

21. Another Solution • E1  E2 = {2, 3, 4, 5, 6}  P(E1  E2) = 5 / 6

22. Example • In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male.

23. Solution • P(nurse or male) = P(nurse) + P(male)  P(male nurse) = (8/13) + (3/13)  (1/13) = 10/13

24. Example A single card is drawn at random from an ordinary deck of cards. Find the probability that it is either an ace or a red card. Solution

25. More than two events

26. Can You see why?

27. Example • If one card is drawn from an ordinary deck of cards, find the probability of getting the following. • A king or a queen or a jack. • A club or a heart or a spade. • A king or a queen or a diamond.

28. Solution (a) • P("king") = 4/52 = 1/13 , P("queen") = 1/13, P("jack") = 1/13 • We note that the events “king, queen and jack” are mutually exclusive. Thus • P("king or a queen or a jack") = P("king") + P("queen") + P("jack") = 3/13

29. Solution(b) • P("club") = P("heart") = P("spade") = 13/52 = ¼ • We note that the events are mutually exclusive, Thus P("club or a heart or a spade") = P("club")+P("heart")+ P("spade") = ¾

30. Solution (C) • P(king) = P(queen) =1/13 • P (diamond) = ¼ • We note that the events are NOTmutually exclusive, Thus, we still have to get P(king and diamond) = 1/52 P(queen and diamond) = 1/52

31. Solution (C) So, P(king or queen or diamond) = P(king)+P(queen)+ P(diamond) - P(king and diamond) - P(queen and diamond) = 4/52 + 4/52 + 13/52 – 1/52 -1/52 = 19/52

32. Example A box consists of 1000 rivets. 50 rivets with type A defect 32 rivets with type B defect 18 rivets with type C defect 7 rivets with type A and B defects 5 rivets with type A and C defects 4 rivets with type B and C defects 2 rivets with type A, B and C defects What is the probability that a rivet picked from the box will have: 1. Type A or Type B defect or both? 2. At least one of the three types of defects? 3. No defects?

33. Solution

34. Solution

35. Dependent and Independent Events • Two events are independent if the occurrence of one does not affect the probability of the other one’s occurring. • Otherwise, the two events are dependent.

36. Note • Here the two events need not be subsets of the same sample space.

37. Examples of Independent Events • Flipping a coin and getting a head and rolling a die and getting a 2. • Rolling one die and getting a 1 and rolling another die and getting a 6. • Drawing a card and getting a king, replacing it, and drawing another card and getting a queen.

38. Examples of Dependent Events • Drawing a card from a deck, not replacing it, and then drawing another card. • Studying well and getting high grades 

39. Multiplication Rule for Independent Events • Let E1 and E2 be independent events, then • The multiplication rules can be used to find the probability of two or more events that occur in sequence.

40. Example • A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 2 on the die.

41. Solution • Let E1 be the event of flipping a coin and getting a head. • Let E2 be the event of rolling a die and getting a 2. • E1 and E2 are independent events.

42. Example • An urn contains 3 red balls, 2 blue balls, and 5 white balls. A ball is selected and its color noted. Then it is replaced. A second ball is selected and its color noted. Find the probability of each of these. (a) Selecting 2 blue balls. (b) Selecting a blue ball and then a white ball. (c) Selecting a red ball and then a blue ball.

43. Solution • Note that all events are independent, (a) P(blue and blue) = (2/10)(2/10) = 2/50 (b) P(blue and white) = (2/10)(5/10) = 1/10 (c) P(red and blue) = (3/10)(2/10) = 3/50

44. Example • A game is played by drawing four cards from an ordinary deck and replacing each card after it is drawn. Find the probability that at least one ace is drawn.

45. Solution • Let E be the event of drawing no aces. Then

46. Example Approximately 9% of men have a type of color blindness that prevents them from distinguishing between red and green. If 3 men are selected at random, find the probability that all of them will have this type of red-green color blindness. Solution Let C denote red-green color blindness. Then

47. Example

48. Multiplication Rule for Dependent Events • When two events A and B are dependent, then the probability of both occurring is P(A and B ) = P(A).P(B\A)

49. Conditional probability where, P(B\A) is the conditional probability of B, i.e. the probability of occurring of B given that A has occurred already.

50. Note • The given rule here of conditionalprobability is the second rule of the multiplication rules. • The multiplication rules are given in two cases according to whether the events are independent or dependent.