Probability and Counting Techniques


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1 Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each of these, choice 2 in m 2 ways; for each of these, choice 3 in m 3 ways; and so forth. Then the task can be performed in m 1 m 2 m 3 m t ways. Exercise 1. How many different twoletter words (including nonsense words) can be formed when repetition of letters is allowed? When repetition is NOT allowed? Exercise 2. How many license plates consisting of two letters followed by four digits are possible? Exercise 3. How many license plates consisting of two letters followed by four digits are possible if letters cannot be repeated? Exercise 4. How many license plates consisting of two letters followed by four digits are possible if digits cannot be related? 1
2 Exercise 5. How many license plates consisting of two letters followed by four digits are possible if both letters and digits cannot be repeated? Exercise 6. How many ways can five people be arranged in a line for a group picture? Exercise 7. Toss a coin six times and observe the sequence of heads or tails that results. How many different sequences are possible? Exercise 8. Six houses in a row are each to be painted with one of the colors red, blue, green, and yellow. In how many different ways can the houses be painted so that no two adjacent houses are of the same color? Exercise 9. How many threedigit odd numbers can be formed using the digits 1, 2, 3, 4, 5, 6, and 7? Combinations and Permutations A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words (strings of letters) of two distinct letters can be formed from the letters {a, b, c}. 2
3 Factorial Combinations (Order does not matter!) Combination Formula: The number of combinations of n objects taken r at a time is Exercise 10. a) C(100, 2) C(n, r) = n! (n r)!r! b) C(5, 5) Exercise 11. The board of directors of a corporation has 10 members. In how many ways can they choose a committee of 3 board members to negotiate a merger? 3
4 Exercise 12. How many different selections of two books can be made from a set of nine books? Permutations (Order matters!) Permutation Formula: The number of permutations of n objects taken r at a time is P (n, r) = n(n 1)(n 2)(n 3) (n r + 1) OR P (n, r) = n! (n r)! Exercise 13. Eight horses are entered in a race in which a first, second, and third prize will be awarded. Assuming no ties, how many different outcomes are possible? Exercise 14. A club has 10 members. In how many ways can they choose a slate of four officers, consisting of a president, vice president, secretary, and treasurer? Exercise 15. How many ways can you arrange 5 of 10 books on a shelf? 4
5 Exercise 16. Suppose that you own 10 sweaters and are going on a trip. How many ways can you select six of them to leave at home? Exercise 17. Of the 20 applicants for a job, 4 will be selected for intensive interviews. In how many ways can the selection be made? Exercise 18. A poker hand consists of 5 cards selected from a deck of 52 cards. a) How many different poker hands are there? b) How many different poker hands consist entirely of aces and kings? 5
6 c) How many different poker hands consist entirely of clubs? d) How many different poker hands consist entirely of red cards? Exercise 19. In how many ways can five mathematics books and four novels be placed on a bookshelf if the mathematics books must stay together? Solving a Permutation Problem with Like Objects Exercise 20. How many different passwords can be made using all the letters in the word Mississippi? 6
7 Exercise 21. (You Try!) How many different passwords can be made using all the letters in the word Massachusetts? Counting Problems Exercise 22. An urn contains 25 numbered balls, of which 15 are red and 10 are white. A sample of 3 balls is to be selected. a) How many different samples are possible? b) How many samples contain all red balls? c) How many samples contain 1 red balls and 2 white balls? 7
8 d) How many samples contain at least 2 red balls? Exercise 23. A package contains 100 fuses, of which 10 are defective. A sample of 3 fuses is selected at random. a) How many different samples are there? b) How many of the samples contain all defective fuses? c) How many samples contain 1 defective fuse and 2 good ones? 8
9 d) How many of the samples contain at least 2 defective fuse? Exercise 24. A student is required to work exactly 6 problems from a 10problem exam and must work exactly 3 of the first 4 problems. In how many ways can the six problems be chosen? Exercise 25. A committee has four male and five female members. In how many ways can a subcommittee consisting of two males and two females be selected? Experiements, Outcomes, Sample Spaces, and Events Definition 1. A sample space is a set of all possible outcomes of an experiment. 9
10 Exercise 26. An experiment consists of flipping a coin once. Find the sample space. Exercise 27. An experiment consist of flipping a coin twice. Find the sample space. Exercise 28. An experiment consists of flipping a coin three times. Find the sample space. Exercise 29. An experiment consists of rolling a single die. Find the sample space. Exercise 30. An experiment consists of rolling a green die and a red die. Find the sample space. 10
11 Exercise 31. An experiment consists of answering four truefalse questions with either a T or an F. Find the sample space. Exercise 32. An experiment consists of drawing a card from an ordinary deck of cards and noting its suit, replacing that card and drawing another, again noting the suit. Describe the outcomes, and find the sample space of the experiment. Exercise 33. An experiment consists of rolling a die twice, each time recording a 0 if the result is an odd number and the number if the result is an even number. Describe the outcomes of this experiment, and find the sample space. 11
12 Exercise 34. A twostage experiment consists of first selecting a bowl from the two shown below and noting its label (a or b) and then selecting a ball from that bowl and noting its color (R is red, W is white, and G is green). Describe the outcomes of this experiment, and find the sample space. Exercise 35. Of the 4 rats in a cage in a psychology laboratory, 3 are untrained and 1 is trained. A rat is removed from the cage, and it is noted whether or not it is trained, and then it is put into another cage. Two more rats are removed and treated in the same way. Draw a tree diagram for this experiment, and find the sample space. Exercise 36. Videocassette recorder (VCR) tapes produced on an assembly line are either acceptable or defective. An experiment consists of checking tapes one after another until either a defective tape has been found of three tapes have been checked. Once a tape has been checked, 12
13 it is set aside and not checked again. Draw a tree diagram to represent the experiment, and find the sample space. Exercise 37. A box contains 1 red, 1 white, and 2 green balls. An experiment consists of drawing balls in succession without replacement and noting the color of each until a red ball is drawn. a) Draw a tree diagram for this experiment. b) How many outcomes are in the sample space? Definition 2. An event is a subset of a sample space of an experiment. Exercise 38. Suppose and experiment consists of tossing a coin three times and observing the sequence of heads and tails. Determine the event E = exactly two heads. 13
14 Exercise 39. Suppose that we have two urns  call them urn I and urn II  each containing red balls and white balls. An experiment consists of selecting an urn and then selecting a ball from that urn and noting its color. a) What is a suitable sample space for this experiment? b) Describe the event urn I is selected as a subset of the sample space. Exercise 40. An experiment consists of tossing a coin four times and observing the sequence of heads and tails. a) What is the sample space of this experiment? b) Determine the event E 1 = more heads than tails occur. c) Determine the event E 2 = the first toss is a head. 14
15 d) Determine the event E 1 E 2. Exercise 41. A political poll surveys a group of people to determine their income levels and political affiliations. People are classified as either low, middle, or upperlevel income and as either Democrat, Republican, or Independent. a) Find the sample space corresponding to the poll. b) Determine the event E 1 = Independent. c) Determine the event E 2 = low income and not Independent. d) Determine the event E 3 = neither upper income nor Independent. 15
16 Definition 3. Suppose E and F are events in a sample space S. We say that E and F are mutually exclusive (or disjoint) provided that E F =. Exercise 42. Let S = {a, b, c, d, e, f, g} be a sample space, and let E = {a, b, c}, F = {e, f, g}, and G = {c, d, f}. a) Are E and F mutually exclusive? b) Are F and G mutually exclusive? c) Are F G and E mutually exclusive? Exercise 43. An experiment consists of selecting a car at random from a college parking lot and observing the color and make. Let E be the event the car is red, F be the event the car is a Chevrolet, G be the event the car is a green Ford, and H be the event the car is black or a Chrysler. (a) Which of the following pairs of events are mutually exclusive? (i) E and F (ii) F and G (iii) F and H (iv) E and G (b) Describe each of the following events: (i) E F (ii) E 16
17 (iii) G (iv) E G Assignment of Probabilities Definition 4. Experiments in which each outcome has the same probability are said to be experiments with equally likely outcomes. Exercise 44. Experiment consists of flipping a coin two times. Find probability of every outcome in the sample space. Exercise 45. A bowl contains 3 balls: 1 red, 1 white, and 1 green. An experiment consists of selecting a ball, noting its color, and then flipping a coin and noting the result, heads or tails. Find the number of outcomes. If the outcomes are equally likely, what weight should be assigned to each? 17
18 Definition 5. If an experiment with sample space S has equally likely outcomes, then for any event E the probability of E is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements in E and S, respectively. Note: Probability is always a number from 0 to 1. Impossible events always have probability 0 and certain events have probability 1. Exercise 46. Roll a single die. What is the probability that it lands on an odd number? Exercise 47. An experiment consists of rolling a red die and a green die and noting the result of each roll. Find the probability that the sum is 8. Definition 6. InclusionExclusion Principle Let E and F be any events. Then P (E F ) = P (E) + P (F ) P (E F ) In particular, if E and F are mutually exclusive, then P (E F ) = P (E) + P (F ) 18
19 Exercise 48. Let E and F be events in a sample space S with P (E) =.65, P (F ) =.4, and P (E F ) =.3. Find P (E F ). Exercise 49. Joe feels that the probability of getting an A in history is.7, the probability of getting an A in psychology is.8, and the probability of getting an A in history or psychology is.9. What is the probability that he will get an A in both subjects? Exercise 50. A sound system is such that when it is used, the microphone malfunctions with probability.1, the speakers malfunction with probability.05, and both malfunction with probability.01. a) What is the probability that neither the microphone nor the speakers malfunction? 19
20 b) What is the probability that either the microphone malfunctions or the speakers malfunction, but not both? Exercise 51. Suppose you are asked to choose a whole number between 1 and 13, inclusive. a) What is the probability that it is odd? b) What is the probability that it is even? c) What is the probability that it is a multiple of 3? d) What is the probability that it is odd or a multiple of 3? Exercise 52. Let E and F be events for which P (E) =.6, P (F ) =.5, and P (E F ) =.4. Find (a) P (E F ) 20
21 (b) P (E F ) Exercise 53. Observe twochild families. Describe the sample space for counting the number of boys, and assign probabilities to each outcome. Exercise 54. Probability distribution for coin toss. Exercise 55. Probability distribution for roll of a die. 21
22 Exercise 56. An experiment with outcomes s 1, s 2, s 3, s 4, s 5, s 6 probability table: is described by the following Outcome Probability s 1.05 s 2.25 s 3.05 s 4.01 s 5.63 s 6.01 Let E = {s 1, s 2 } and F = {s 3, s 5, s 6 }. a) Determine P (E) and P (F ). b) Determine P (E ). c) Determine P (E F ). d) Determine P (E F ). Converting between Odds and Probabilities If the odds in favor of the event E occurring are a to b, then P (E) = a a + b If P (E) = p, then the odds in favor of E are found by reducing the fraction p to the form a, 1 p b where a and b are integers having no common divisor. Then the odds in favor of E are. a to b 22
23 Exercise 57. What are the odds of obtaining a three when rolling a die. Exercise 58. The probability of obtaining a sum of eight or more when rolling a pair of dice is. What are the odds of obtaining a sum of eight or more? Exercise 59. Four people are running for class president: Liz, John, Sue, and Tom. probabilities of John, Sue, and Tom winning are.18,.23, and.31, respectively. The (a) What is the probability of Liz winning? (b) What is the probability that a boy wins? (c) What is the probability that Tom loses? (d) What are the odds that Sue loses? 23
24 (e) What are the odds that a girl wins? (f) What are the odds that John wins? Exercise 60. An urn contains 8 white balls and two green balls. A sample of 3 balls is selected at random. a) What is the total number of possible samples? b) How many samples contain all white balls? c) What is the probability of selecting only white balls? d) What is the probability of selecting two white balls and one green ball? e) What is the probability that the sample contains at least one green ball? 24
25 Definition 7. Let E be any event, E its complement. Then. P (E) = 1 P (E ) Exercise 61. A toy manufacturer inspects boxes of toys before shipment. Each cox contains 10 toys. The inspection procedure consists of randomly selecting three toys from the box. If any are defective, the box is not shipped. Suppose that a given box has two defective toys. What is the probability that it will be shipped? Exercise 62. Suppose that we toss a coin 5 times. a) What is the probability of getting exactly 3 heads? b) What is the probability of at most one head occurring? c) What is the probability that both heads and tails occur? 25
26 Calculating Probabilities of Events Exercise 63. An urn contains five red balls and four white balls. A sample of 2 balls is selected at random from the urn. a) What is the probability of selecting only white balls? b) What is the probability of selecting only red balls? 26
27 c) What is the probability that at least one of the balls is red? Exercise 64. A factory produces fuses, which are packaged in boxes of 10. Three fuses are selected at random from each box for inspection. The box is rejected if at least one of these three fuses is defective. What is the probability that a box containing five defective fuses will be rejected? 27
28 Exercise 65. A softball team has 6 aluminum bats and 9 magnesium bats. Three bats are selected at random from the bat rack. a) Write down all the possible outcomes. b) What is the probability that all 3 are aluminum? c) What is the probability that at least one bat is aluminum? d) What is the probability that both aluminum and magnesium bats are selected? Find the solution using two methods. 28
29 Exercise 66. (You Try!) A bowl contains 6 red ball and 4 green balls. A sample of 3 balls is selected at random. a) Write down all the possible outcomes. b) What is the probability that both red and green balls are obtained? You should obtain the same solution using two methods. Method I is when you obtain the solution by directly computing the number of possible outcomes. Method II is when you use complement rule and subtract the unwanted outcomes. 29
30 Exercise 67. A bag contains nine tomatoes, of which one is rotten. A sample of three tomatoes is selected at random. a) How many tomatoes are in the bag? b) How many tomatoes in the bag are rotten? c) How many tomatoes in the bag are not rotten? d) Draw a bag indicating the numbers of rotten and not rotten tomatoes. Under the bag write down how many you are selecting. e) What is the total number of possible samples? 30
31 f) Write down all the possible outcomes? g) What is the probability that a sample contains the rotten tomato? Recall the Complement Rule: P (E) = 1 P (E ) Exercise 68. (Birthday Problem) A group of five people is to be selected at random. What is the probability that two or more of them have the same birthday? (Assume that each of the 365 days in a year is an equally likely birthday) Exercise 69. A basketball team has four players. What is the probability that at least two of them were born on the same day of the week? 31
32 Exercise 70. A high school astrology club has 3 members. What is the probability that two or more members have the same zodiac sign? (Note: there are 12 zodiac signs) Exercise 71. There were 16 presidents of the Continental Congresses from 1774 to Each of the five students in a seminar in American history chooses one of these on which to do a report. If all presidents are equally likely to be chosen, calculate the probability that at least two students choose the same president. Exercise 72. (Probability distribution for the number of girls in twochild families) Observe twochild families. Describe the sample space for counting the number of girls, and assign probabilities to each outcome. (We will need to consider gender and birth order). 32
33 Exercise 73. A couple decides to have five children. a) Describe the sample space for counting the number of boys. b) What is the probability they will have exactly one boy? c) What is the probability they will have exactly three boys? d) What is the probability they will have one boy or three boys? e) What is the probability that they will have more boys than girls? 33
34 Exercise 74. A couple decides to have four children. a) Describe the sample space for counting the number of girls. b) What is the probability they will have exactly one girl? c) What is the probability they will have at most one girl? d) What is the probability they will have at least one child of each sex? 34
35 Calculating Probabilities of Events and Conditional Probability Exercise 75. Two coins are selected at random from 5 coins: 3 dimes and 2 quarters. What is the probability that both are dimes? Exercise 76. A group of 6 students  4 males and 2 females  selects 2 students at random to make a report. Find the probability that two females are selected. Exercise 77. Three cards are selected at random from a deck of 52 cards. What is the probability that all three are spades? Exercise 78. Four cards are selected at random from a deck of 52 cards. What is the probability that all 4 are aces? Exercise 79. A fair coin is flipped 4 times. What is the probability that both heads and tails occur? 35
36 Exercise 80. Two distinct numbers are selected at random from the set {1, 2, 3, 4, 5, 6}. What is the probability that their product is an even number? (Hint: First find the probability that the product is odd) Exercise 81. Two fair dice are rolled, and the sum is noted. Find the probability that the sum is less than 6. Exercise 82. Suppose that a certain mathematics class contains 26 students. Of these, 14 are economics majors, 15 are firstyears students, and 7 are neither. Suppose that a person is selected at random from the class. a) What is the probability that the person is both an economics major and a firstyear student? b) Suppose we are given the additional information that the person selected is a firstyear student. What is the probability that he or she is also an economics majors? 36
37 Definition 8. In case of equally likely outcomes, the conditional probability of E given F, written P (E F ) is [number of outcomes in E F ] P (E F ) = [number of outcomes in F] provided that [number of outcomes in F] 0. Exercise 83. Suppose that we have an urn containing 3 red balls, 2 green balls, and 4 white balls, and a ball is selected at random and its color is noted. Assuming that each ball is equally likely to be selected, i.e. each ball has probability 1 of being selected, find 9 a) P (red ball is selected) = b) P (green ball is selected) = c) P (white ball is selected) = Now suppose that we are given additional information that the ball selected is not white. Since only 5 of the 9 balls are not white, we may think of the sample space as being limited to those 5 outcomes. With this information, find the following probabilities: a) P (red ball selected not white) = 37
38 b) P (green ball selected not white) = c) P (white ball selected not white) = Exercise 84. Two balls are selected at random from an urn containing two white balls and three red balls. What is the conditional probability that both balls are white given that at least one of them is white? Exercise 85. Out of 250 thirdgrade boys, 120 played baseball, 140 played soccer, and 50 played both. Find the probability that a boy chosen at random a) did not play either sport. b) played exactly one sport. 38
39 c) played soccer but not baseball. d) played soccer, given that he played baseball. e) played baseball, given that he did not play soccer. f) did not play soccer, given that he did not play baseball. Definition 9. The conditional probability of E given F, written P (E F ) is provided that P (F ) 0 P (E F ) = P (E F ) P (F ) 39
40 Exercise 86. Let A and B be events such that P (A) =.7, P (B) =.6, and P (A B) =.4 Find the following: a) P (A B) b) P (B A) c) P (A B ) d) P (A B) Exercise 87. The proportion of individuals in a certain city earning more than $35, 000 per year is.25. The proportion of individuals earning more than $35, 000 and having a college degree is.10. Suppose that a person is randomly chosen and he turns out to be earning more than $35, 000. What is the probability that he is a college graduate? 40
41 Conditional Probability and Tree Diagrams Exercise 88. Let E and F be events in a sample space S. Suppose P (E) = 5 8, P (F ) = 3 8, and P [(E F ) ] = 0. Find P (E F ) and P (F E). Exercise 89. Two fair dice are rolled, and the number on the uppermost faces are noted. a) What is the probability that exactly one die shows a 4 given that the sum of the numbers is 6? b) What is the probability that the sum of the numbers is 6 given that exactly one die shows a 3? c) What is the probability that the sum of the numbers is 6 given that exactly one die shows a 4? 41
42 Exercise 90. There are 4 men and 5 women on a committee. A subcommittee of 3 people is selected at random. Find the probability that all are male given that all are the same sex. Trees Exercise 91. Consider an experiment consisting of two steps. Firs a box is selected at random from a set of two boxes labeled a and b, and then an urn is selected at random from the chosen box. Box a contains 3 urns, labeled A, B, and C, and box b contains 2 urns, labeled D and E. Draw a tree diagram to represent the outcomes of this experiment and assign conditional probabilities to the tree to reflect the probabilities of the possible results at each stage. 42
43 Exercise 92. Consider the threestage experiment which uses the boxes, urns, and colored balls (see the picture below) and which proceeds as follows: Select a box, then select an urn, then select a ball from that urn, and note the color of the ball. Suppose that all selections are random. a) Form a tree diagram for this experiment, and compute the probabilities for all outcomes. b) Find the probabilities of these events: F = {a red ball is drawn }, G = {a white ball is drawn}, H = {a green ball is drawn }. 43
44 Exercise 93. A box has: 5 white balls, 6 red balls, and 3 yellow balls. Choose 2 balls without replacement. a) Draw a tree diagram and label all the probabilities. b) Find the probability that the first ball is red and second ball is white. Exercise 94. Toyota makes 55% of its Prius in US and 45% in Japan. 10% of USmade cars are defective, and 2% of Japanmade cars are defective. If my car is defective, what is the probability it was made in US? (Hint: draw a tree) 44
45 Exercise 95. Suppose that 60% of the voters in US are Democrats and 40% are Republicans. 85% of democrats voted for Kerry and 90% of republicans voted for Bush. What is the probability that a Bush voter is a republican? Exercise 96. Of people with TB, 98% have a positive reaction and 2% have a negative reaction. Of people free of TB, 99% have a negative reaction and 1% have a positive reaction. From a large population of which 2 per 10,000 persons have TB, a person is selected at random and given a skin test, which turns out to be positive. What is the probability that the person has TB? 45
46 Exercise 97. There are 9 apples in a bag; 5 are red and 4 are yellow. Two apples are selected at random, one after another without replacement, and the color of each is noted. a) Find the probability that the second is red. b) Find the probability that at least 1 is red. Exercise 98. Repeat the experiment of Exercise 10, but in this case place the first apple back in the bag before selecting the second apple. Find the same two probabilities. 46
47 Conditional Probability and Independence Exercise 99. Of the students at a certain college, 50% regularly attend the football games, 30% are firstyear students, and 40% are upperclass students who do not regularly attend football games. Suppose that a student is selected at random. a) What is the probability that the person both is a firstyear student and regularly attends football games? b) What is the conditional probability that the person regularly attends football games given that he is a firstyear student? c) What is the conditional probability that the person is firstyear student given that he regularly attends football games? Definition 10. Let E and F be events. We say E and F are independent provided that P (E F ) = P (E)P (F ) 47
48 Exercise 100. An experiment consists of observing the outcome of two consecutive rolls of a die. Let E and F be the events E = the first roll is a 3 F = the second roll is a 6 Show that these events are independent. Exercise 101. Suppose that we toss a coin three times and record the sequence of heads and tails. Let E be the event at most one head occurs and F the event both heads and tails occur. Are E and F independent? Exercise 102. Suppose that a family has four children. Let E be the event at most one boy and F the event at least one child of each sex. Are E and F independent? 48
49 Exercise 103. A company manufactures stereo components. Experience shows that defects in manufacture are independent of one another. Quality control studies reveal that 2% of CD players are defective, 3% of amplifiers are defective, 7% of speakers are defective A system consists of a CD player, an amplifier, and two speakers. What is the probability that the system is not defective? Exercise 104. The table below shows number of degrees awarded (in thousands) in the US in a recent year. Bachelor s Master s Professional Doctorate Total Female Male Total If a degree recipient is chosen at random, what is the probability that the chosen person a) is a female recipient of a doctorate? 49
50 b) received a doctorate, given that the person chosen is a woman? c) is a man, given that the person received a professional or doctorate degree? d) Are the events female and doctoral recipient independent? How do you know? Exercise 105. An urn contains 10 red balls and 15 white balls (balls are numbered). A sample of four balls is selected at random form the urn. (a) What is the probability that at least one ball is white? 50
51 (b) If a sample of four balls contains at least one white ball, what is the probability that all the balls in the sample are white? Exercise 106. Events E and F are independent in a sample space S with P (E) = 0.3 and P (F ) = 0.6. Find P (E F ). Exercise 107. Kim has a strong first serve; whenever it is good (that is, in) she wins the point 75% of the time. Whenever her second serve is good, she wins the point 50% of the time. Sixty percent of her first serves and 75% of her second serves are good. (If the first serve is good, Kim does not do the second serve). (a) Draw the tree diagram. (b) What is the probability that Kim wins the point when she serves? 51
52 (c) If Kim wins a service point, what is the probability that her first serve was good? Exercise 108. An urn contains three balls numbered 1,2, and 3. Balls are drawn one at a time without replacement until the sum of the numbers drawn is four or more. Find the probability of stopping after exactly two balls are drawn. 52
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