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Chapter 3 Probability Larson/Farber 4th ed 1

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Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule 3.3 The Addition Rule 3.4 Additional Topics in Probability and Counting Larson/Farber 4th ed 2

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Section 3.1 Basic Concepts of Probability Larson/Farber 4th ed 3

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Section 3.1 Objectives Identify the sample space of a probability experiment Identify simple events Use the Fundamental Counting Principle Distinguish among classical probability, empirical probability, and subjective probability Determine the probability of the complement of an event Use a tree diagram and the Fundamental Counting Principle to find probabilities Larson/Farber 4th ed 4

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Probability Experiments Probability experiment An action, or trial, through which specific results (counts, measurements, or responses) are obtained. Outcome The result of a single trial in a probability experiment. Sample Space The set of all possible outcomes of a probability experiment. Event Consists of one or more outcomes and is a subset of the sample space. Larson/Farber 4th ed 5

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Probability Experiments Probability experiment: Roll a die Outcome: {3} Sample space: {1, 2, 3, 4, 5, 6} Event: {Die is even}={2, 4, 6} Larson/Farber 4th ed 6

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Example: Identifying the Sample Space A probability experiment consists of tossing a coin and then rolling a six-sided die. Describe the sample space. Larson/Farber 4th ed 7 Solution: There are two possible outcomes when tossing a coin: a head (H) or a tail (T). For each of these, there are six possible outcomes when rolling a die: 1, 2, 3, 4, 5, or 6. One way to list outcomes for actions occurring in a sequence is to use a tree diagram.

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Solution: Identifying the Sample Space Larson/Farber 4th ed 8 Tree diagram: H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6 The sample space has 12 outcomes: {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

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Simple Events Simple event An event that consists of a single outcome. e.g. “Tossing heads and rolling a 3” {H3} An event that consists of more than one outcome is not a simple event. e.g. “Tossing heads and rolling an even number” {H2, H4, H6} Larson/Farber 4th ed 9

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Example: Identifying Simple Events Determine whether the event is simple or not. You roll a six-sided die. Event B is rolling at least a 4. Larson/Farber 4th ed 10 Solution: Not simple (event B has three outcomes: rolling a 4, a 5, or a 6)

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Fundamental Counting Principle If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m*n. Can be extended for any number of events occurring in sequence. Larson/Farber 4th ed 11

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Example: Fundamental Counting Principle You are purchasing a new car. The possible manufacturers, car sizes, and colors are listed. Manufacturer: Ford, GM, Honda Car size: compact, midsize Color: white (W), red (R), black (B), green (G) How many different ways can you select one manufacturer, one car size, and one color? Use a tree diagram to check your result. Larson/Farber 4th ed 12

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Solution: Fundamental Counting Principle There are three choices of manufacturers, two car sizes, and four colors. Using the Fundamental Counting Principle: 3 ∙ 2 ∙ 4 = 24 ways Larson/Farber 4th ed 13

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Types of Probability Classical (theoretical) Probability Each outcome in a sample space is equally likely. Larson/Farber 4th ed 14

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Example: Finding Classical Probabilities 1.Event A: rolling a 3 2.Event B: rolling a 7 3.Event C: rolling a number less than 5 Larson/Farber 4th ed 15 Solution: Sample space: {1, 2, 3, 4, 5, 6} You roll a six-sided die. Find the probability of each event.

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Solution: Finding Classical Probabilities 1.Event A: rolling a 3 Event A = {3} Larson/Farber 4th ed 16 2.Event B: rolling a 7 Event B= { } (7 is not in the sample space) 3.Event C: rolling a number less than 5 Event C = {1, 2, 3, 4}

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Types of Probability Empirical (statistical) Probability Based on observations obtained from probability experiments. Relative frequency of an event. Larson/Farber 4th ed 17

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Example: Finding Empirical Probabilities A company is conducting an online survey of randomly selected individuals to determine if traffic congestion is a problem in their community. So far, 320 people have responded to the survey. What is the probability that the next person that responds to the survey says that traffic congestion is a serious problem in their community? Larson/Farber 4th ed 18 ResponseNumber of times, f Serious problem123 Moderate problem115 Not a problem 82 Σf = 320

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Solution: Finding Empirical Probabilities Larson/Farber 4th ed 19 ResponseNumber of times, f Serious problem123 Moderate problem115 Not a problem82 Σf = 320 eventfrequency

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Law of Large Numbers As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event. Larson/Farber 4th ed 20

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Types of Probability Subjective Probability Intuition, educated guesses, and estimates. e.g. A doctor may feel a patient has a 90% chance of a full recovery. Larson/Farber 4th ed 21

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Example: Classifying Types of Probability Classify the statement as an example of classical, empirical, or subjective probability. Larson/Farber 4th ed 22 Solution: Subjective probability (most likely an educated guess) 1.The probability that you will be married by age 30 is 0.50.

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Example: Classifying Types of Probability Classify the statement as an example of classical, empirical, or subjective probability. Larson/Farber 4th ed 23 Solution: Empirical probability (most likely based on a survey) 2.The probability that a voter chosen at random will vote Republican is 0.45.

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3.The probability of winning a 1000-ticket raffle with one ticket is. Example: Classifying Types of Probability Classify the statement as an example of classical, empirical, or subjective probability. Larson/Farber 4th ed 24 Solution: Classical probability (equally likely outcomes)

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Range of Probabilities Rule Range of probabilities rule The probability of an event E is between 0 and 1, inclusive. 0 ≤ P(E) ≤ 1 Larson/Farber 4th ed 25 [ ] ImpossibleUnlikely Even chance LikelyCertain

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Complementary Events Complement of event E The set of all outcomes in a sample space that are not included in event E. Denoted E ′ (E prime) P(E ′) + P(E) = 1 P(E) = 1 – P(E ′) P(E ′) = 1 – P(E) Larson/Farber 4th ed 26 E ′ E

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Example: Probability of the Complement of an Event You survey a sample of 1000 employees at a company and record the age of each. Find the probability of randomly choosing an employee who is not between 25 and 34 years old. Larson/Farber 4th ed 27 Employee agesFrequency, f 15 to to to to to and over42 Σf = 1000

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Solution: Probability of the Complement of an Event Use empirical probability to find P(age 25 to 34) Larson/Farber 4th ed 28 Employee agesFrequency, f 15 to to to to to and over42 Σf = 1000 Use the complement rule

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Example: Probability Using a Tree Diagram A probability experiment consists of tossing a coin and spinning the spinner shown. The spinner is equally likely to land on each number. Use a tree diagram to find the probability of tossing a tail and spinning an odd number. Larson/Farber 4th ed 29

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Solution: Probability Using a Tree Diagram Tree Diagram: Larson/Farber 4th ed 30 HT H1H2H3H4H5H6H7H8 T1T2T3T4T5T6T7T8 P(tossing a tail and spinning an odd number) =

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Example: Probability Using the Fundamental Counting Principle Your college identification number consists of 8 digits. Each digit can be 0 through 9 and each digit can be repeated. What is the probability of getting your college identification number when randomly generating eight digits? Larson/Farber 4th ed 31

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Solution: Probability Using the Fundamental Counting Principle Each digit can be repeated There are 10 choices for each of the 8 digits Using the Fundamental Counting Principle, there are 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 = 10 8 = 100,000,000 possible identification numbers Only one of those numbers corresponds to your ID number Larson/Farber 4th ed 32 P(your ID number) =

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Section 3.1 Summary Identified the sample space of a probability experiment Identified simple events Used the Fundamental Counting Principle Distinguished among classical probability, empirical probability, and subjective probability Determined the probability of the complement of an event Used a tree diagram and the Fundamental Counting Principle to find probabilities Larson/Farber 4th ed 33

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Section 3.2 Conditional Probability and the Multiplication Rule Larson/Farber 4th ed 34

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Section 3.2 Objectives Determine conditional probabilities Distinguish between independent and dependent events Use the Multiplication Rule to find the probability of two events occurring in sequence Use the Multiplication Rule to find conditional probabilities Larson/Farber 4th ed 35

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Conditional Probability The probability of an event occurring, given that another event has already occurred Denoted P(B | A) (read “probability of B, given A”) Larson/Farber 4th ed 36

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Example: Finding Conditional Probabilities Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. (Assume that the king is not replaced.) Larson/Farber 4th ed 37 Solution: Because the first card is a king and is not replaced, the remaining deck has 51 cards, 4 of which are queens.

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Example: Finding Conditional Probabilities The table shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ, given that the child has the gene. Larson/Farber 4th ed 38 Gene Present Gene not presentTotal High IQ Normal IQ Total

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Solution: Finding Conditional Probabilities There are 72 children who have the gene. So, the sample space consists of these 72 children. Larson/Farber 4th ed 39 Of these, 33 have a high IQ. Gene Present Gene not presentTotal High IQ Normal IQ Total

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Independent and Dependent Events Independent events The occurrence of one of the events does not affect the probability of the occurrence of the other event P(B | A) = P(B) or P(A | B) = P(A) Events that are not independent are dependent Larson/Farber 4th ed 40

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Example: Independent and Dependent Events 1.Selecting a king from a standard deck (A), not replacing it, and then selecting a queen from the deck (B). Larson/Farber 4th ed 41 Dependent (the occurrence of A changes the probability of the occurrence of B) Solution: Decide whether the events are independent or dependent.

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Example: Independent and Dependent Events Decide whether the events are independent or dependent. 2.Tossing a coin and getting a head (A), and then rolling a six-sided die and obtaining a 6 (B). Larson/Farber 4th ed 42 Independent (the occurrence of A does not change the probability of the occurrence of B) Solution:

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The Multiplication Rule Multiplication rule for the probability of A and B The probability that two events A and B will occur in sequence is P(A and B) = P(A) ∙ P(B | A) For independent events the rule can be simplified to P(A and B) = P(A) ∙ P(B) Can be extended for any number of independent events Larson/Farber 4th ed 43

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Example: Using the Multiplication Rule Two cards are selected, without replacing the first card, from a standard deck. Find the probability of selecting a king and then selecting a queen. Larson/Farber 4th ed 44 Solution: Because the first card is not replaced, the events are dependent.

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Example: Using the Multiplication Rule A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6. Larson/Farber 4th ed 45 Solution: The outcome of the coin does not affect the probability of rolling a 6 on the die. These two events are independent.

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Example: Using the Multiplication Rule The probability that a particular knee surgery is successful is Find the probability that three knee surgeries are successful. Larson/Farber 4th ed 46 Solution: The probability that each knee surgery is successful is The chance for success for one surgery is independent of the chances for the other surgeries. P(3 surgeries are successful) = (0.85)(0.85)(0.85) ≈ 0.614

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Example: Using the Multiplication Rule Find the probability that none of the three knee surgeries is successful. Larson/Farber 4th ed 47 Solution: Because the probability of success for one surgery is The probability of failure for one surgery is 1 – 0.85 = 0.15 P(none of the 3 surgeries is successful) = (0.15)(0.15)(0.15) ≈ 0.003

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Example: Using the Multiplication Rule Find the probability that at least one of the three knee surgeries is successful. Larson/Farber 4th ed 48 Solution: “At least one” means one or more. The complement to the event “at least one successful” is the event “none are successful.” Using the complement rule P(at least 1 is successful) = 1 – P(none are successful) ≈ 1 – = 0.997

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Example: Using the Multiplication Rule to Find Probabilities More than 15,000 U.S. medical school seniors applied to residency programs in Of those, 93% were matched to a residency position. Seventy-four percent of the seniors matched to a residency position were matched to one of their top two choices. Medical students electronically rank the residency programs in their order of preference and program directors across the United States do the same. The term “match” refers to the process where a student’s preference list and a program director’s preference list overlap, resulting in the placement of the student for a residency position. (Source: National Resident Matching Program) Larson/Farber 4th ed 49 (continued)

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Example: Using the Multiplication Rule to Find Probabilities 1.Find the probability that a randomly selected senior was matched a residency position and it was one of the senior’s top two choices. Larson/Farber 4th ed 50 Solution: A = {matched to residency position} B = {matched to one of two top choices} P(A) = 0.93 and P(B | A) = 0.74 P(A and B) =P(A)∙P(B | A) = (0.93)(0.74) ≈ dependent events

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Example: Using the Multiplication Rule to Find Probabilities 2.Find the probability that a randomly selected senior that was matched to a residency position did not get matched with one of the senior’s top two choices. Larson/Farber 4th ed 51 Solution: Use the complement: P(B′ | A) = 1 – P(B | A) = 1 – 0.74 = 0.26

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Section 3.2 Summary Determined conditional probabilities Distinguished between independent and dependent events Used the Multiplication Rule to find the probability of two events occurring in sequence Used the Multiplication Rule to find conditional probabilities Larson/Farber 4th ed 52

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Section 3.3 Addition Rule Larson/Farber 4th ed 53

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Section 3.3 Objectives Determine if two events are mutually exclusive Use the Addition Rule to find the probability of two events Larson/Farber 4th ed 54

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Mutually Exclusive Events Mutually exclusive Two events A and B cannot occur at the same time Larson/Farber 4th ed 55 A B AB A and B are mutually exclusive A and B are not mutually exclusive

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Example: Mutually Exclusive Events Decide if the events are mutually exclusive. Event A: Roll a 3 on a die. Event B: Roll a 4 on a die. Larson/Farber 4th ed 56 Solution: Mutually exclusive (The first event has one outcome, a 3. The second event also has one outcome, a 4. These outcomes cannot occur at the same time.)

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Example: Mutually Exclusive Events Decide if the events are mutually exclusive. Event A: Randomly select a male student. Event B: Randomly select a nursing major. Larson/Farber 4th ed 57 Solution: Not mutually exclusive (The student can be a male nursing major.)

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The Addition Rule Addition rule for the probability of A or B The probability that events A or B will occur is P(A or B) = P(A) + P(B) – P(A and B) For mutually exclusive events A and B, the rule can be simplified to P(A or B) = P(A) + P(B) Can be extended to any number of mutually exclusive events Larson/Farber 4th ed 58

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Example: Using the Addition Rule You select a card from a standard deck. Find the probability that the card is a 4 or an ace. Larson/Farber 4th ed 59 Solution: The events are mutually exclusive (if the card is a 4, it cannot be an ace) 4♣ 4♥4♥ 4♦4♦ 4♠ A♣ A♥A♥ A♦A♦ A♠ 44 other cards Deck of 52 Cards

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Example: Using the Addition Rule You roll a die. Find the probability of rolling a number less than 3 or rolling an odd number. Larson/Farber 4th ed 60 Solution: The events are not mutually exclusive (1 is an outcome of both events) Odd Less than three Roll a Die

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Solution: Using the Addition Rule Larson/Farber 4th ed 61 Odd Less than three Roll a Die

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Example: Using the Addition Rule The frequency distribution shows the volume of sales (in dollars) and the number of months a sales representative reached each sales level during the past three years. If this sales pattern continues, what is the probability that the sales representative will sell between $75,000 and $124,999 next month? Larson/Farber 4th ed 62 Sales volume ($)Months 0–24, ,000–49, ,000–74, ,000–99, ,000–124, ,000–149, ,000–174, ,000–199,9991

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Solution: Using the Addition Rule A = monthly sales between $75,000 and $99,999 B = monthly sales between $100,000 and $124,999 A and B are mutually exclusive Larson/Farber 4th ed 63 Sales volume ($)Months 0–24, ,000–49, ,000–74, ,000–99, ,000–124, ,000–149, ,000–174, ,000–199,9991

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Example: Using the Addition Rule A blood bank catalogs the types of blood given by donors during the last five days. A donor is selected at random. Find the probability the donor has type O or type A blood. Larson/Farber 4th ed 64 Type OType AType BType ABTotal Rh-Positive Rh-Negative Total

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Solution: Using the Addition Rule Larson/Farber 4th ed 65 The events are mutually exclusive (a donor cannot have type O blood and type A blood) Type OType AType BType ABTotal Rh-Positive Rh-Negative Total

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Example: Using the Addition Rule Find the probability the donor has type B or is Rh- negative. Larson/Farber 4th ed 66 Solution: The events are not mutually exclusive (a donor can have type B blood and be Rh-negative) Type OType AType BType ABTotal Rh-Positive Rh-Negative Total

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Solution: Using the Addition Rule Larson/Farber 4th ed 67 Type OType AType BType ABTotal Rh-Positive Rh-Negative Total

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Section 3.3 Summary Determined if two events are mutually exclusive Used the Addition Rule to find the probability of two events Larson/Farber 4th ed 68

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Section 3.4 Additional Topics in Probability and Counting Larson/Farber 4th ed 69

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Section 3.4 Objectives Determine the number of ways a group of objects can be arranged in order Determine the number of ways to choose several objects from a group without regard to order Use the counting principles to find probabilities Larson/Farber 4th ed 70

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Permutations Permutation An ordered arrangement of objects The number of different permutations of n distinct objects is n! (n factorial) n! = n∙(n – 1)∙(n – 2)∙(n – 3)∙ ∙ ∙3∙2 ∙1 0! = 1 Examples: 6! = 6∙5∙4∙3∙2∙1 = 720 4! = 4∙3∙2∙1 = 24 Larson/Farber 4th ed 71

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Example: Permutation of n Objects The objective of a 9 x 9 Sudoku number puzzle is to fill the grid so that each row, each column, and each 3 x 3 grid contain the digits 1 to 9. How many different ways can the first row of a blank 9 x 9 Sudoku grid be filled? Larson/Farber 4th ed 72 Solution: The number of permutations is 9!= 9∙8∙7∙6∙5∙4∙3∙2∙1 = 362,880 ways

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Permutations Permutation of n objects taken r at a time The number of different permutations of n distinct objects taken r at a time Larson/Farber 4th ed 73 ■ where r ≤ n

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Example: Finding nPr Find the number of ways of forming three-digit codes in which no digit is repeated. Larson/Farber 4th ed 74 Solution: You need to select 3 digits from a group of 10 n = 10, r = 3

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Example: Finding nPr Forty-three race cars started the 2007 Daytona 500. How many ways can the cars finish first, second, and third? Larson/Farber 4th ed 75 Solution: You need to select 3 cars from a group of 43 n = 43, r = 3

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Distinguishable Permutations The number of distinguishable permutations of n objects where n 1 are of one type, n 2 are of another type, and so on Larson/Farber 4th ed 76 ■ where n 1 + n 2 + n 3 +∙∙∙+ n k = n

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Example: Distinguishable Permutations A building contractor is planning to develop a subdivision that consists of 6 one-story houses, 4 two- story houses, and 2 split-level houses. In how many distinguishable ways can the houses be arranged? Larson/Farber 4th ed 77 Solution: There are 12 houses in the subdivision n = 12, n 1 = 6, n 2 = 4, n 3 = 2

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Combinations Combination of n objects taken r at a time A selection of r objects from a group of n objects without regard to order Larson/Farber 4th ed 78 ■

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Example: Combinations A state’s department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire four of the bidding companies. How many different combinations of four companies can be selected from the 16 bidding companies? Larson/Farber 4th ed 79 Solution: You need to select 4 companies from a group of 16 n = 16, r = 4 Order is not important

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Solution: Combinations Larson/Farber 4th ed 80

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Example: Finding Probabilities A student advisory board consists of 17 members. Three members serve as the board’s chair, secretary, and webmaster. Each member is equally likely to serve any of the positions. What is the probability of selecting at random the three members that hold each position? Larson/Farber 4th ed 81

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Solution: Finding Probabilities There is only one favorable outcome There are ways the three positions can be filled Larson/Farber 4th ed 82

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Example: Finding Probabilities You have 11 letters consisting of one M, four Is, four Ss, and two Ps. If the letters are randomly arranged in order, what is the probability that the arrangement spells the word Mississippi? Larson/Farber 4th ed 83

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Solution: Finding Probabilities There is only one favorable outcome There are distinguishable permutations of the given letters Larson/Farber 4th ed letters with 1,4,4, and 2 like letters

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Example: Finding Probabilities A food manufacturer is analyzing a sample of 400 corn kernels for the presence of a toxin. In this sample, three kernels have dangerously high levels of the toxin. If four kernels are randomly selected from the sample, what is the probability that exactly one kernel contains a dangerously high level of the toxin? Larson/Farber 4th ed 85

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Solution: Finding Probabilities The possible number of ways of choosing one toxic kernel out of three toxic kernels is 3 C 1 = 3 The possible number of ways of choosing three nontoxic kernels from 397 nontoxic kernels is 397 C 3 = 10,349,790 Using the Multiplication Rule, the number of ways of choosing one toxic kernel and three nontoxic kernels is 3 C 1 ∙ 397 C 3 = 3 ∙ 10,349,790 3 = 31,049,370 Larson/Farber 4th ed 86

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Solution: Finding Probabilities The number of possible ways of choosing 4 kernels from 400 kernels is 400 C 4 = 1,050,739,900 The probability of selecting exactly 1 toxic kernel is Larson/Farber 4th ed 87

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Section 3.4 Summary Determined the number of ways a group of objects can be arranged in order Determined the number of ways to choose several objects from a group without regard to order Used the counting principles to find probabilities Larson/Farber 4th ed 88

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