Chapter 11: Probability and Counting Techniques


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1 Chapter 11: Probability and Counting Techniques Diana Pell Section 11.1: The Fundamental Counting Principle 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? Sections : 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? 2
3 How many words (strings of letters) of two distinct letters can be formed from the letters {a, b, c}. 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) 3
4 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? 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? 4
5 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? 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. 5
6 a) How many different poker hands are there? b) How many different poker hands consist entirely of aces and kings? 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 6
7 Exercise 20. How many different passwords can be made using all the letters in the word Mississippi? Exercise 21. (You Try!) How many different passwords can be made using all the letters in the word Massachusetts? Exercise 22. 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? 7
8 Exercise 23. 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? d) How many samples contain at least 2 red balls? 8
9 Exercise 24. (You Try!) A fourperson crew for the international space station is to be chosen from a candidate pool of 10 Americans and 12 Russians. How many different crews are possible if there must be at least two Russians? Section 11.3: Basic Concepts of Probability Definition 1. A sample space is a set of all possible outcomes of an experiment. Exercise 25. An experiment consists of flipping a coin once. Find the sample space. Exercise 26. An experiment consist of flipping a coin twice. Find the sample space. Exercise 27. An experiment consists of flipping a coin three times. Find the sample space. 9
10 Exercise 28. An experiment consists of rolling a single die. Find the sample space. Exercise 29. An experiment consists of rolling a green die and a red die. Find the sample space. Definition 2. An event is a subset of a sample space of an experiment. Exercise 30. 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. Exercise 31. 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? 10
11 b) Describe the event urn I is selected as a subset of the sample space. Definition 3. Experiments in which each outcome has the same probability are said to be experiments with equally likely outcomes. Exercise 32. Experiment consists of flipping a coin two times. Find probability of every outcome in the sample space. Definition 4. 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 33. A single die is rolled. Find the probability of getting a) A 2. 11
12 b) A number less than 5. Exercise 34. Roll a single die. What is the probability that it lands on an odd number? Complement Rule: P (E) = 1 P (E ) Exercise 35. Of the next 32 trials on the docket in a county court, 5 are homicides, 12 are drug offenses, 6 are assaults, and 9 are property crimes. If jurors are assigned to trials randomly, a) what s the probability that a given juror won t get a homicide case? 12
13 b) what s the probability that a juror gets assigned to a case that isn t a drug offense? Exercise 36. (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 37. 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? Exercise 38. In a random sample of 500 people, 210 had type O blood, 223 had type A, 51 had type B, and 16 had type AB. Set up a frequency distribution and find the probability that a randomly selected person from the general population has 13
14 a) Type O blood. b) Type A or B blood. c) Neither type A nor type O blood. d) A blood type other than AB. Section 11.4: Tree Diagrams, Tables, and Sample Spaces Exercise 39. Use a tree diagram to find the sample space for the genders of three children in a family. Exercise 40. (You Try!) A soda machine dispenses both Coke and Pepsi products, in both 12 ounce cans and 20ounce bottles. For each brand, it has a regular cola, diet cola, and lemonlime drink. Use a tree diagram to find the sample space for the experiment of choosing one drink at random from this machine. 14
15 Exercise 41. A coin is flipped, and then a die is rolled. Use a tree diagram to find the probability of getting heads on the coin and an even number on the die. Exercise 42. (You Try!) In order to collect information for a student survey, a researcher classifies students according to eye color (blue, brown, green), gender (male, female), and class rank (freshman, sophomore). A folder for each classification is then made up (e.g., freshman/female/green eyes). Find the sample space for the folders using a tree diagram. If a folder is selected at random, find the probability that a) It includes students with blue eyes. b) It includes students who are female. c) It includes students who are male freshmen. 15
16 Section 11.5: Probability Using Permutations and Combinations Exercise 43. Stacy has the option of selecting three books to read for a humanities course. The suggested book list consists of 10 biographies and five current events books. She decides to pick the three books at random. Find the probability that all three books will be current events books. Exercise 44. (You Try!) There are 12 women and 8 men in a seminar course. If the professor chooses fiveperson groups at random, what is the probability that the first group chosen will consist of all women? Exercise 45. What is the probability of getting 4 aces when drawing 5 cards from a standard deck of 52 cards? 16
17 Exercise 46. (You Try!) Suppose the deck of cards in the example above has all 32 cards with numbers less than 10 removed, so that only 10s, jacks, queens, kings, and aces remain. Now what is the probability of getting 4 aces when drawing 5 cards? Exercise 47. A combination lock has 40 numbers on it, from zero to 39. Find the probability that if the combination to unlock it consists of three numbers, it will contain the numbers 1, 2, and 3 in some order. Assume that numbers cannot be repeated in the combination. (It s interesting to note that a combination lock should really be called a permutation lock since the order of the numbers is important when you are unlocking the lock.) Exercise 48. (You Try!) A different?permutation? lock has letters from A through L on it, and the combination consists of four letters with no repeats. What is the probability that the combination is I, J, K, and L in some order? 17
18 Exercise 49. A store has six different fitness magazines and three different news magazines. If a customer buys three magazines at random, find the probability that the he ll pick two fitness magazines and one news magazine. Exercise 50. (You Try!) Find the probability that the customer in the example above picks at least two fitness magazines. Exercise 51. The list of potential parolees at a monthly parole hearing consists of eight drug offenders, five violent offenders, and two convicted of property crimes. I d surely like to think that parolees aren t chosen at random, but if this particular board chooses three parolees randomly, find the probability that a) All three are drug offenders. 18
19 b) Two of the three are property offenders. c) All three are violent offenders. d) One of each type of offender is paroled. e) Two are drug offenders and one is a violent offender. 19
20 Section 11.6: Odds and Expectations 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 Exercise 52. What are the odds of obtaining a three when rolling a die. Exercise 53. 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 54. 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? 20
21 (c) What is the probability that Tom loses? (d) What are the odds that Sue loses? (e) What are the odds that a girl wins? (f) What are the odds that John wins? Exercise 55. (You Try!) A card is drawn from a standard deck of 52 cards. (a) Find the odds in favor of getting an ace. (b) Find the odds against getting an ace. 21
22 Another concept related to probability is expectation, or expected value. Expected value is used to determine the result that would be expected over the long term in some sort of gamble. Here s the key thing to remember as we study expected value: it only makes sense for events that have numerical outcomes. Consider probability distribution for a random variable X. Table 1: Probability distribution Outcome Probability x 1 p 1 x 2 p 2 x 3 p 3.. x k p k Recall that the expected value of X, denoted by E(X), is defined to be E(X) = x 1 p 1 + x 2 p x k p k Exercise 56. When a single die is rolled, find the expected value of the outcome. Exercise 57. The prize in a raffle is a flatscreen TV valued at $350, and 1,000 tickets are sold. What s the expected value if you buy 1 ticket? 22
23 Exercise 58. You pay a dollar to roll two dice. If you roll 5 or 6, you get your dollar back plus two more just like it. If not, you get nothing and like it. Find the expected value of playing this game 100 times. Exercise 59. (You Try!) On a roulette wheel, there are 38 slots, 18 of which are colored red. If you bet $5 on red and win, you get $10 back. If red doesn t come up, you lose your $5. Find the expected value of playing the game 100 times. 23
24 Exercise 60. Suppose that the following game is proposed by a friend: A fair die is to be rolled, and if an outcome of 2, 3, 4, or 5 occurs, then your friend will pay you $1.50. If either a 1 or a 6 occurs, then you pay your friend an amount in dollars equal to the outcome. That is, you pay $1.00 if a 1 occurs and $6.00 if a 6 occurs. If you play the game many times, how much gain (or loss) should you expect (i.e. find the expected value)? Exercise 61. You pay $2 to play. You pick a card at random from a standard deck of cards. If you pick the ace of spades, you win $13. Any other ace wins $10. Any face card (jack, queen, or king) wins $5. All other cards win nothing. Do you want to play? Why or why not? Calculate your expected winnings. Note: The expected value of a completely fair game is 0. 24
25 Exercise 62. You pay $2 to buy a lottery ticket. There are 1, 000 tickets in all. There is one first place prize of $400, five second place prizes or $200 and ten third place prizes of $50. Let X represent net winnings. a) Find probability distribution of X. b) Find E(X). c) Is this a fair game? Why? Section 11.7: The Addition Rules for Probability Definition 5. Two events are mutually exclusive if they cannot both occur at the same time. That is, the events have no outcomes in common. Exercise 63. In drawing cards from a standard deck, determine whether the two events are mutually exclusive or not. a) Drawing a 4, drawing a 6. b) Drawing a 4, drawing a heart. 25
26 Exercise 64. (You Try!) If student government picks students at random to win free books for a semester, determine whether the two events are mutually exclusive or not. a) The winner is a sophomore or a business major. b) The winner is a junior or a senior. Definition 6. (Addition Rule I) When two events A and B are mutually exclusive, the probability that A or B will occur is P (A B) = P (A) + P (B) Exercise 65. A restaurant has three pieces of apple pie, five pieces of cherry pie, and four pieces of pumpkin pie in its dessert case. If a customer selects at random one kind of pie for dessert, find the probability that it will be either cherry or pumpkin. Exercise 66. (You Try!) A liberal arts math class contains 7 freshmen, 11 sophomores, 5 juniors, and 2 seniors. If the professor randomly chooses one to present a homework problem at the board, find the probability that it s either a junior or senior. Exercise 67. A card is drawn from a standard deck. Find the probability of getting an ace or a queen. 26
27 Exercise 68. (You Try!) At a political rally, there are 20 Republicans, 13 Democrats, and 6 Independents. If a person is selected at random, find the probability that he or she is either a Democrat or an Independent. Exercise 69. A card is drawn from a deck. Find the probability that it is either a club, a diamond, or a heart. Exercise 70. (You Try!) In rolling two dice, find the probability that the sum is 2, 3, or 4. Definition 7. (Addition Rule II) When two events A and B are NOT mutually exclusive, the probability that A or B will occur is P (A B) = P (A) + P (B) P (A B) 27
28 Exercise 71. A single card is drawn from a standard deck of cards. Find the probability that it s a king or a club. Exercise 72. (You Try!) A card is drawn from an ordinary deck. Find the probability that it is a heart or a face card. Exercise 73. Two dice are rolled. Find the probability of getting doubles or a sum of 6. Exercise 74. (You Try!) When two dice are rolled, find the probability that both numbers are more than three, or that they differ by exactly two. 28
29 Exercise 75. 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 76. In a hospital there are eight nurses and five physicians. Seven nurses and three physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male. Exercise 77. (You Try!) In one class, there are 15 freshmen and 10 sophomores. Six of the freshmen are education majors and four of the sophomores are education majors. If a student is selected at random, find the probability that the student is a sophomore or an education major. 29
30 Definition 8. Two events A and B are independent if the fact that A occurs has no effect on the probability of B occurring. 1) Rolling a die and getting a 6, and then rolling a second die and getting a 3. 2) Drawing a card from a deck and getting a queen, replacing it, and drawing a second card and getting a queen. Definition 9. Two events A and B are dependent if the outcome of A has some effect on the probability of B occurring. 1) Drawing a card from a deck, not replacing it, and then drawing a second card. 2) Parking in a noparking zone and getting a parking ticket. Multiplication Rule 1 When two events A and B are independent, the probability of both occurring is P (A B) = P (A) P (B) Exercise 78. A coin is flipped and a die is rolled. Find the probability of getting heads on the coin and a 4 on the die. Exercise 79. As part of a psychology experiment on perception and memory, colored balls are picked from an urn. The urn contains three red balls, two green balls, and five white balls. A ball is picked and its color is noted. Then it is replaced. A second ball is picked and its color is noted. Find the probability of each of these. a) Picking two green balls. b) Picking a green ball and then a white ball. 30
31 c) Picking a red ball and then a green ball. Exercise 80. (You Try!) As part of a card trick, a card is drawn from a deck and replaced; then a second card is drawn. Find the probability of getting a queen and then an ace. Exercise 81. Three cards are drawn from a deck. After each card is drawn, its denomination and suit are noted and it s mixed back into the deck before the next card is drawn. Find the probability of getting a) Three kings. b) Three clubs. Exercise 82. According to a study done by the Princeton Review in 2012, 86 percent of collegebound students indicated that financial aid would be?very necessary? for them to attend college. If four collegebound students were chosen at random, find the probability that all four would rate financial aid as very necessary. 31
32 Conditional Probability Exercise 83. 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? Definition 10. The probability that a second event B occurs given that a first event A has occurred can be found by dividing the probability that both events occurred by the probability that the first event has occurred. The formula is P (B A) = P (A B) P (A) Exercise 84. Suppose that your professor goes stark raving mad and chooses your final grade from A, B, C, D, F, or Incomplete totally at random. Find the probability of getting an A given that you get a letter grade higher than D. 32
33 Exercise 85. (You Try!) A group of patients in a blind drug trial is assigned numbers from 1 through 8. The even numbers get an experimental drug, while the odd numbers get a placebo. If Eleanor is one of the patients, what s the probability that she s getting the experimental drug given that she wasn t assigned 1, 2, or 3? Exercise 86. Hate crimes are defined to be crimes in which the victim is targeted because of one or more personal characteristics, such as race, religion, or sexual orientation. The table below lists the motivation for certain hate crimes as reported by the FBI for Motivation Crimes against persons Crimes against property Crimes agains society Race 2, Religion Sexual Orientation Total 4,009 2, a) Find the probability that a hate crime was racially motivated given that it was a crime against persons. b) Find the probability that a hate crime was against property given that it was motivated by the victim s sexual orientation. 33
34 c) (You Try!) A crime was motivated by either race or religion given that it was a crime against society. d) (You Try!) A crime was against persons given that it was motivated by religion or sexual orientation. Section 11.9: The Binomial Distribution Experiments with just two outcomes are called binomial trials. Here are some examples of binomial trials. 1. Toss a coin and observe the outcome, heads or tails. 2. Administer a drug to a sick individual and classify the reaction as effective or ineffective. 3. Manufacture a light bulb and classify it as nondefective or defective. The outcomes of a binomial trial are usually called success or failure. We will denote the probability of success by p and probability of failure by q. Since binomial trial has only two outcomes we have p + q = 1, or q = 1 p X  success F  failure p = P (X) and q = P (F ) and q = 1 p (1) Repeat experiment n times. (2) Outcome is X or F. (3) Repeated trials are independent. 34
35 What is the probability of k successes and n k failures? ( ) n P (X = K) = p k q n k k Exercise 87. Find the probability of obtaining exactly two heads when tossing a fair coin three times. Exercise 88. Find the probability of obtaining exactly 17 heads when tossing a coin 20 times. Exercise 89. A plumbingsupplies manufacturer produces faucet washers, which are packaged in boxes of 300. Quality control studies have shown that 2% of the washers are defective. What is the probability that a box of washers contains exactly 9 defective washers? 35
36 Exercise 90. Each time at bat the probability that a baseball player gets a hit is.3. He comes up to a bat four times in a game. Assume that his times at bat are independent trials. Find the probability that he gets a) exactly two hits b) at least two hits Exercise 91. A survey found that 33% of people earning between 30, 000and75,000 said that they were very happy. If 6 people who earn between 30, 000and75,000 are selected at random, find the probability that at most 2 would consider themselves very happy. 36
37 Exercise 92. The recovery rate for a certain cattle disease is 25%. If 40 cattle are afflicted with the disease, what is the probability that exactly 10 will recover? Exercise 93. In a 20question truefalse test, what is the probability of answering exactly 18 questions correctly just by guessing? Exercise 94. A professor who intends to bring her briefcase to the office each morning forgets it onequarter of the time. Assume that forgetting the briefcase is a Binomial trial, and find the probability that she forgets it at least twice a week (5 days). 37
38 PRACTICE Exercise 95. Of five physical therapists that work at a rehab center, three have master s degrees and two have doctorates. Each therapist is equally likely to be assigned to a patient on any given visit. If Tom has five sessions scheduled in the next two weeks, find the probability that a) He gets a therapist with a doctorate twice. b) He gets a therapist with a doctorate less than two times. Exercise 96. In a recent survey, 2% of the people surveyed said that they would keep their current job if they won a multimilliondollar lottery. If 20 people are chosen randomly, find the probability that 3, 4, or 5 of them would keep their job. 38
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