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Chapter 3 Probability Larson/Farber 4th ed

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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

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

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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

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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

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

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Example: Identifying the Sample Space A probability experiment consists of tossing a three coins. Describe the sample space. Larson/Farber 4th ed Solution: { HHH, HHT, HTT, HTH, HTT, THH, THT, TTH, TTT }

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Solution: Identifying the Sample Space Larson/Farber 4th ed Tree diagram: The sample space has 8 outcomes: {HHH, HHT, HTT, HTH, THH, THT, TTH, TTT}

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Simple Events Simple event An event that consists of a single outcome. e.g. “You randomly select a card from standard deck. Event C is selecting a four of hearts” An event that consists of more than one outcome is not a simple event. e.g. “A computer is used to randomly select a number between 1 and 200. Event B is selecting a number less than 33.” Larson/Farber 4th ed

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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

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Example: Fundamental Counting Principle Do #14 on page 142. Larson/Farber 4th ed

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Solution: Fundamental Counting Principle There are three choices of salad, six main dishes, and four desserts. Using the Fundamental Counting Principle: 3 ∙ 6 ∙ 4 = 72 ways Larson/Farber 4th ed

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

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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 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 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

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Example: Finding Empirical Probabilities The number of voters (in millions) according to age. Larson/Farber 4th ed Age of Votersf 18 - 205.8 21 - 248.5 25 - 3421.7 35 - 4427.7 45 - 6451.7 65 and older26.7 142.1

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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

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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

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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 [ ] 00.51 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 E ′ E

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Example: Probability of the Complement of an Event Back to our voter example: #45 - 48 Larson/Farber 4th ed Age of Votersf 18 - 205.8 21 - 248.5 25 - 3421.7 35 - 4427.7 45 - 6451.7 65 and older26.7 142.1

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Solution: Probability of the Complement of an Event Use empirical probability to find P(age 25 to 34) = p(E) Larson/Farber 4th ed Use the complement rule, find p( age not 25 to 34) = p(E’) Age of Votersf 18 - 205.8 21 - 248.5 25 - 3421.7 35 - 4427.7 45 - 6451.7 65 and older26.7 142.1

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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

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