1. Chapter # 4 Probability and Counting Rules

2. 4-1 Sample Spaces and Probability 4-2 Addition Rules for Probability 4-3 Multiplication Rules & Conditional Probability 4-4 Counting Rules 4-5 Probability and Counting Rules Introduction

3.  Probability  Probability can be defined as the chance of an event occurring.

4.  A probability experiment is a chance process that leads to well-defined results called outcomes.  An outcome is the result of a single trial of a probability experiment.  A sample space is the set of all possible outcomes of a probability experiment.  An event consists of outcomes. S OME D EFINITIONS

5. S OME S AMPLE S PACES Roll a die ExperimentSample Space Toss a coin Answer a true/false question S={True, False} Toss two coinsS={HH,H HT, TH, TT} S={Head, Tail }

6.  A tree diagram is a device consisting of line segments emanating from a starting point and also from the outcome point.  It is used to determine all possible outcomes of a probability experiment.

7. G ENDER OF C HILDREN Example 4-3: three Find the sample space and tree diagram for the gender of the children if a family has three children. Use (B for boy) and (G for girl).

8. Solution : B G B G B G B G B G B G B G BBB BBG BGB BGG GBB GBG GGB GGG

9.  An event consists of outcomes of a probability Experiment. For example : event Simple event is an event with one outcome. Compound event is an event with containing more than one outcome S = { 1, 2, 3, 4, 5, 6 } A = { 6 } B = Odd no. = { 1, 3, 5 } E = Even no. = { 2, 4, 6 } Simple event Compound event

10.  There are three basic interpretations of probability:   Classical probability  Empirical probability  Subjective probability

11. Classical probability

12.  Classical probability  Classical probability uses sample spaces to determine the numerical probability that an event will happen and assumes that all outcomes in the sample space are equally likely to occur.  Equally Likely Events are events that have the same probability of occurring

13. There are four basic probability rules:   First Rule: 0 ≤ P(E) ≤ 1  Second Rule: If an event E cannot occur,then the probability is 0.  Third Rule: If an event E is certain, then the probability of E is 1.  Fourth Rule: ∑ p = 1 Probability Rules

14. If a family has three children, find the probability that two of the three children are girls. Step 1 : Sample Space: S ={BBB,BBG, BGB, BGG, GBB,GBG,GGB,GGG} Step 2 : k={BGG, GBG, GGB} P(K)= = 3/8 The probability of having two of three children being girls is 3/8. G ENDER OF C HILDREN Example 4-6: Solution :

15. Example 4-8: Solution : When a single die is rolled, find the probability of getting a 9. Sample Space: 1, 2, 3, 4, 5, 6 P(9) = 0/6 = 0 R OLLING A D IE Example 4-9: When a single die is rolled,what is the probability of getting a number less than 7 ?. Sample Space: 1, 2, 3, 4, 5, 6 P(no. less than 7)= 6/6 = 1 Solution : Second Rule Fourth Rule Note: This PowerPoint is only a summary and your main source should be the book.

16. Complement of Event Rolling a die and getting a 4Getting a 1, 2, 3, 5, or 6 Selecting a letter of the alphabet and getting a vowel Selecting a month and getting a month that begins with a J Selecting a day of the week and getting a weekday Event Getting a consonant (assume y is a consonant) Getting February, March, April, May, August, September, October, November, or December Getting Saturday or Sunday Example 4-10: Find the complements of each event. The complement of an event E, denoted by, is the set of outcomes in the sample space that are not included in the outcomes of event E.

17. P (E) P (S)

18. Rule for Complementary Events Example 4-11: If the probability that a person lives in an industrialized country of the world is, find the probability that a person does not live in an industrialized country.

19. Empirical Probability

20. Empirical probability(relative) relies on actual experience to determine the likelihood of outcomes.

21. In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities: Example 4-13: TypeFrequency A22 B5 AB2 O21 Total 50 a. A person has type O blood. b. A person has type A or type B blood. c. A person has neither type A nor type O blood. d. A person does not have type AB blood.

22. Subjective probability

23. Subjective probability uses a probability value based on an educated guess or estimate, employing opinions and inexact information. (based on: the person’s experience and evaluation of solution) Examples: weather forecasting, predicting outcomes of sporting events *EX: The probability that a bus will be in an accident on a specific run is about 6%.

24. A ball is chosen at random from a box containing 5 black, 8 red and 7 yellow balls. Find the probability that it is : a) red b) yellow c) not black

25. Addition Rules for Probability Addition Rules for Probability

26. *** Addition Rules *** mutually exclusive events if they cannot occur at the same time (i.e., they have no outcomes in common) not mutually exclusive events Can occur at the same time (i.e., outcomes in common not mutually exclusive) P(A∩B)= 0

27. ** If P(A) = 0.3, P(B) = 0.4, and A,B are mutually exclusive events, find P(A and B). a) 0 b) 1 c) 0.12 d) 0.7

28. Example 4-15: R OLLING A D IE Determine which events are mutually exclusive and which are not, when a single die is rolled. a. Getting an odd number and getting an even number Getting an odd number: 1, 3, or 5 Getting an even number: 2, 4, or 6 Mutually Exclusive b. Getting a 3 and getting an odd number Getting a 3: 3 Getting an odd number: 1, 3, or 5 Not Mutually Exclusive

29. c. Getting an odd number and getting a number less than 4 Getting an odd number: 1, 3, or 5 Getting a number less than 4: 1, 2, or 3 Not Mutually Exclusive d. Getting a number greater than 4 and getting a number less than 4 Getting a number greater than 4: 5 or 6 Getting a number less than 4: 1, 2, or 3 Mutually Exclusive

30. 1.Determine which events are mutually exclusive. a) Select a student in your college: The student is in the second year and the student is a math major. b) Select a child: The child has black hair and the child has black eyes. c) Roll a die: Get a number greater than 2 and get a multiple of 3. d) Roll a die: Get a number greater than 3 and get a number less than 3.

31. Example 4-17: A box contains 3 glazed doughnuts, 4 jelly doughnuts, and 5 chocolate doughnuts. If a person selects a doughnut at random,find the probability that it is either a glazed doughnut or a chocolate doughnut. Solution :

32. Example 4-19:

33. Example 4-21: 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. Solution : StaffFemalesMalesTotal Nurses718 Physicians325 Total10313

34. Multiplication Rules

35. *** Multiplication Rules *** Independent Events If the fact that A occurs does not affect the probability of B occurring. EX: Rolling a die and getting a 6, then rolling another die and getting 3 Dependent Events The outcome of the first event affects the outcome of the second event. EX: Having high grades and getting a scholarship (ثقافة).

36. S ELECTING A C OLORED B ALL Example 4-25: An urn contains 3 red balls, 2blue 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.

37. M ALE C OLOR B LINDNESS Example 4-27: Approximately 9% of men have a type of color blindness (عمى ألوان)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 P(C and C and C) = P(C). P(C). P(C) = (0.09)(0.09)(0.09) = 0.000729

38. E XAMPLE 4-28: U NIVERSITY C RIME At a university in western Pennsylvania, there were 5 burglaries reported in 2003, 16 in 2004, and 32 in 2005. If a researcher wishes to select at random two burglaries to further investigate, find the probability that both will have occurred in 2004. Solution :

39. E XAMPLE 4-29: H OMEOWNER ’ S AND A UTOMOBILE I NSURANCE World Wide Insurance Company found that 53% of the residents of a city had homeowner’s insurance (Hتأمين صاحب البيت ) with the company.Of these clients,27% also had automobile insurance (Aتأمين سيارة) with the company.If a resident is selected at random,find the probability that the resident has both homeowner’s and automobile insurance with World Wide Insurance Company. Solution :

40. E XAMPLE 4-31: S ELECTING C OLORED B ALLS Box 1 contains 2 red balls and 1 blue ball. Box 2 contains 3 blue balls and 1 red ball. A coin is tossed. If it falls heads up,box1 is selected and a ball is drawn. If it falls tails up,box 2 is selected and a ball is drawn. Find the probability of selecting a red ball. Box 2 Box 1

41. Solution : Coin Box 1 Box 2 Red Blue

42. ** contains 20% defective transistors, contains 30% defective transistors, and contains 50% defective transistors. A die is rolled. If the number that appears is greater than 3, a transistor is selected from 1. If the number is less than 3, a transistor is selected from 2. If the number is 3, a transistor is selected from. Find the probability of selecting a defective transistor. a) 0.028 b) 1 c) 0.283 d) 0.03 ** A die is rolled. What is the probability that the number rolled is greater than 2 and even number? 1/3

43. Conditional Probability

44.   Conditional probability is the probability that the second event B occurs given that the first event A has occurred.

45. E XAMPLE 4-32: S ELECTING C OLORED C HIPS A box contains black chips and white chips. A person selects two chips without replacement. If the probability of selecting a black chip and a white chip is, and the probability of selecting a black chip on the first draw is, find the probability of selecting the white chip on the second draw,given that the first chip selected was a black chip. Solution : Let B=selecting a black chip W=selecting a white chip

46. E XAMPLE 4-34: S URVEY ON W OMEN I N THE M ILITARY A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results of the survey are shown. a. the respondent answered yes, given that the respondent was a female. b. the respondent was a male, given that the respondent answered no. c. the respondent was a female or answered no. Find the probability that

47. E XAMPLE 4-36: A coin is tossed 5 times. Find the probability of getting at least 1 tail ? E=at least 1 tail E= no tail ( all heads) P(E)=1-P(E) P(at least 1 tail)=1- p(all heads) P ROBABILITIES FOR “ AT LEAST ”

48. ** If 63% of children play computer games, and 4 of them are chosen at random, find the probability that all four play computer games. A)0.158 A)2.52 A)1.48 A)0.019 ** If 0.35 of men are smokers and 3 men are selected at random, find the probability that at least one is a smoker. a) 0.043 b) 0.274 c) 0.476 d) 0.725

49. Counting Rules 1- Fundamental Counting Rule 2- Permutation 3- Combination

50.  In a sequence of n events in which the first one has k 1 possibilities and the second event has k 2 and the third has k 3, and so forth, the total number of possibilities of the sequence will be k 1 · k 2 · k 3 · · · k n 1-Fundamental Counting Rule Event 1 Event n …………… k1k1 k2k2 knkn

51. A paint manufacturer wishes to manufacture several different paints. The categories include Color: red, blue, white, black, green, brown, yellow Type: latex, oil Texture: flat, semi gloss, high gloss Use: outdoor, indoor How many different kinds of paint can be made if you can select one color, one type, one texture, and one use? E XAMPLE 4-39:

52. If a menu has a choice of 8 appetizer (مقبلات), 6 main courses (أطباق رئيسية), 5 deserts (حلى), then the sample space for all possible lunch can determined by using: a) The addition rule. b) The combination rule. c) The fundamental counting rule. -------------------------------------------------------------------------- The digits 0,1,2,3,4,5,6,7,8,9 are to be used in a four-digit ID card. How many different cards are possible if repetitions are permitted? a) 16 b) 10000 c) 100000

53.   Permutation is an arrangement of n objects in a specific order using r objects. (Order matters). ------------------------------------------------------------------ 2-Permutation   Factorial is the product of all the positive numbers from 1 to a number.

54. Suppose a business owner has a choice of 5 locations in which to establish her business. She decides to rank each location according to certain criteria, such as price of the store and parking facilities. How many different ways can she rank the 5 locations? Example 4-42:

55. E XAMPLE 4-44: A television news director wishes to use 3 news stories on an evening show One story will be the lead story, one will be the second story, and the last will be a closing story. If the director has a total of 8 stories to choose from, how many possible ways can the program be set up?. Solution :

56.   Combination is a grouping of objects order does not matter. ( selection of distinct objects without regard to order ) 3- 3-Combination

57. E XAMPLE 4-47: How many combinations of 4 objects are there. Taken 2 at a time? Solution : 4c24c2

58. In a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there? Example 4-49: Solution :

59. E XAMPLE 4-51: A box contains 24 transistors, 4 of which are defective. If 4 are sold at random, find the following probabilities : a- Exactly 2 are defective. b-Non is defective. C- All are defective.

60. d-At least 1 is defective

61. ** In a statistics department, there are six teachers, 4 of which are males. If 2 teachers are selected at random, what is the probability that both of them are females? a) 2/3 b) 2/5 c) 1/15 d) 1/3

62. ** How many different ID cards can be made if there are 2 letters followed by 2 digits and none of them can be used more than once? a) 58500

63. how many outcomes are possible if both numbers selected must be even? a) 2

64. E XAMPLE 4-52: A store has 6 TV Graphic magazines and 8 Newstime magazines on the counter. If two customers purchased a magazine, find the probability that one of each magazine was purchased. Solution: P( 1 TV Graphic and 1 Newstime)

65. * A new employee has an option of 5 health care plans, 3 retirement plans and 2 different expense accounts. Find the probability that a person can select one of each option? [5C1.3C1.2C1]/10C3=0.25

66. E XAMPLE 4-53: A combination lock consist of the 26 letters of the alphabet. If a 3- letter combination is needed, find the probability that the combination will consist of the letters ABC in that order.The same letter can be used more than once. Solution :

67. E XAMPLE 4-54: There are 8 married couples in a tennis club. If 1 man and 1 woman are selected at random to plan the summer tournament, find the probability that they are married to each other. Solution : P(they are married to each other )=

68. Revision

69. 1- A store manager wants to display 6 different brands of shampoo in a raw. How many different ways can this be done? a) 120 b) 720 c) 6 d) 36 --------------------------------------------------------------------------------------------------------- 2- If the probability that a person lives in a village is 0.6, what is the probability that a person does not live in a village? a) 0.2 b) 0.4 c) 0.3 d) 0.6

70. 3- If P(A)=0.3, P(B)=0.4 and P(A and B)=0.10, then the events A and B are said to be: a) Mutually exclusive events. b) Not mutually exclusive events. c) Independent events. d) Dependent events. --------------------------------------------------------------------------------------------------------- 4- If P(A)=0.3, P(B)=0.2 and P(A or B)=0.5, then the events A and B are said to be: a) Mutually exclusive events. b) Not mutually exclusive events. c) Independent events. d) Dependent events.

71. 5- A jar contains 3 red marbles, 7 green marbles and 10 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is white? A-0.5 B-0.4 C-0.3 D-0.8 ----------------------------------------------------------------------------------

72. 7- How many different letter arrangements can be made from the word: “sample”? A- 720 B- 820 C- 920 D- 1020 ---------------------------------------------------------------------------------- 8- How many different letter arrangements can be made from the word: “success”? A- 720 B- 620 C- 520 D- 420

73. 9- The sample space for the children gender in a family with three children is (B: boy, G: girl): a) 4 b) 8 c) S={BBB, BBG, BGB, BGG, GBB, GBG, GGB,GGG} d) S={BBG, BGB, BGG, GBB, GBG, GGB} -------------------------------------------------------------------------------------------------------- 10- The statement “The probability that an earthquake will occur in a certain area is 30%”. This is an example of: a) Classical probability. b) Empirical probability. c) Subjective probability.

74. 11- A store has 4 adventure stories (قصص مغامرات) and 5 horror stories (قصص الرعب) on the counter. If two customers purchased a story, find the probability that one of each story was purchased: a) 5/9 b) 4/9 c) 50/100 -------------------------------------------------------------------------------------------------------- 12- Which of these numbers cannot be a probability: a) 0.01 b) 2% c) -0.01

75. 13- The probabilities of the events A and B are: P(A and B)=0.2, P(B│A)=0.3. Find the a) 0.3 b) 0.4 c) 0.5 -------------------------------------------------------------------------------------------------------- 14- What type of probability uses sample space to determine the probability that an event occur? a) Classical probability. b) Empirical probability. c) Subjective probability.

76. 15- A package contains 12 flash memories, 3 of which are defective. If 4 are selected, find the probability of getting 3 defective flash memories? a) 1/55 b) 5/55 c) 28/55 --------------------------------------------------------------------------- 16- How many different ways can a person select 3 cars from 6 cars in a specific order? a) 120 b) 20 c) 66

77. 17- Box A contains 4 red balls and 2 blue ball. Box B contains 2 blue balls and 2 red ball. A die is rolled. If the outcome is an even number a ball is chosen at random from Box A. If the outcome is an odd number a ball is chosen at random from Box B. Find the probability of choosing a red ball. a)7/12 b)1/12 c)5/50

78. 18- How many ways can a person select 6 physics books and 5 math books from 9 physics books and 11 math books? ---------------------------------------------------------------------------

79. 19- What is the probability that a person is a Master and TA? a) 0.309 b) 0.425 c) 1.5 What is the probability that a person is a Bachelor given that he is a Admin? a) 0.425 c) 0.233 c) 0.408 AdminTATotal Bachelor7815 Master231740 Total 302555

80. 20- The probability of any event D is: a) 0 ≤ P(D) ≤ 1 b) -1 < P(D) <1 c) 0 < P(D) ≤ 1 d) 0 ≤ P(D) < 2 --------------------------------------------------------------------------- 21- A die is rolled one time, find the probability of getting number less than or equal 2 or an even number. a) 1 b) 2/3 c) 5/3 d) 4/8

81. 22- When an event is certain, what is its probability? a) 1 b) 0 c) 0.5 --------------------------------------------------------------------------- 23- When an event is impossible, what is its probability? a) 1 b) 0 c) 0.5 --------------------------------------------------------------------------- 24- a married couple has three children, find the probability they are all boys? a) 1/8 b) 3/8

82. 25- The probability that a student has a car is 0.8, and the probability that he has an I-Phone is 0.7, while the probability that he either car or I-Phone is 0.6. Find the probability that he has both. a) 0.9 b) 0.6 c) 0.8

83. 26- d. 27/55

84. 32- A student has to sell 2-book, from a collection of 6-math, 7-art, 4-economic books. How many choices are possible: ** Both books are to be the same subject: a) 40 b) 42 ** The books are to be on different subject: a) 94 b) 90

85. 33 If a coin is tossed 4 times. Find the probability of getting at least 1 head? a) ½ b) 15/16 c) 1/8