Chapter 3 Probability Larson/Farber 4th ed. Chapter Outline 3.1 Basic Concepts of Probability 3.2...
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Transcript of Chapter 3 Probability Larson/Farber 4th ed. Chapter Outline 3.1 Basic Concepts of Probability 3.2...
![Page 1: Chapter 3 Probability Larson/Farber 4th ed. Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule.](https://reader036.fdocuments.net/reader036/viewer/2022081419/56649f325503460f94c4da32/html5/thumbnails/1.jpg)
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
![Page 3: Chapter 3 Probability Larson/Farber 4th ed. Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule.](https://reader036.fdocuments.net/reader036/viewer/2022081419/56649f325503460f94c4da32/html5/thumbnails/3.jpg)
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
![Page 5: Chapter 3 Probability Larson/Farber 4th ed. Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule.](https://reader036.fdocuments.net/reader036/viewer/2022081419/56649f325503460f94c4da32/html5/thumbnails/5.jpg)
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.”
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Fundamental Counting Principle
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.
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Example: Fundamental Counting Principle
Do #14 on page 142.
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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
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Types of Probability
Classical (theoretical) Probability• Each outcome in a sample space is equally likely.
•
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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.
•
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Example: Finding Empirical Probabilities
The number of voters (in millions) according to age.
Larson/Farber 4th ed
Age of Voters f
18 - 20 5.8
21 - 24 8.5
25 - 34 21.7
35 - 44 27.7
45 - 64 51.7
65 and older 26.7
142.1
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Law of Large Numbers
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.
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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.
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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
[ ]0 0.5 1
Impossible UnlikelyEven
chance Likely Certain
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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 Voters f
18 - 20 5.8
21 - 24 8.5
25 - 34 21.7
35 - 44 27.7
45 - 64 51.7
65 and older 26.7
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 Voters f
18 - 20 5.8
21 - 24 8.5
25 - 34 21.7
35 - 44 27.7
45 - 64 51.7
65 and older 26.7
142.1
![Page 24: Chapter 3 Probability Larson/Farber 4th ed. Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule.](https://reader036.fdocuments.net/reader036/viewer/2022081419/56649f325503460f94c4da32/html5/thumbnails/24.jpg)
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