Definition: Event that either A, B, or both occur.Venn Diagram: Shaded area includes all of A and B.
Intersection of Events (A ∩ B)
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Definition: Event that both A and B occur.Venn Diagram: Shaded area includes only the overlap of A and B.
Complement of an Event (Aᶜ)
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Definition: Event that A does not occur.Venn Diagram: Shaded area includes everything outside of A.
Important Relationships
Union (A ∪ B): "Either...or...or both"
Intersection (A ∩ B): "Both...and" or just "and"
Complement (Aᶜ): "Not A"
Extensions to Multiple Events
Union of Three Events (A ∪ B ∪ C):
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Definition: Set of simple events in A, B, or C or any combination of them.
Intersection of Three Events (A ∩ B ∩ C):
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Definition: Collection of simple events common to A, B, and C.
Main topic
4.5. Bayes’ Rule
Key Concepts
Event Definitions
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B: The person selected is a manBᶜ: The person selected is a womanA: The person is colorblindSample Space (S): Consists of both men and womenMutual Exclusivity: Events that cannot occur simultaneously
Decomposition of Event A
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A=(A∩B)∪(A∩Bc)𝑃(𝐴)=𝑃(𝐴∩𝐵)+𝑃(𝐴∩𝐵𝑐)
Law of Total Probability
4.1. Events and the Sample Space
Experiment
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Definition: Process by which an observation or measurement is obtained.
Simple Event
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Definition: Outcome observed on a single repetition of an experiment.
Event
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Definition: Collection of simple events.
Mutually Exclusive Events
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Definition: Two events are mutually exclusive if, when one event occurs, the other cannot, and vice versa.
Sample Space (S)
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Definition: The set of all simple events.
Visuals
Tree Diagram
Venn Diagram
4.2. Calculating Probabilities Using Simple Events
Definition of Probability
Characteristics of Probability
Requirements for Simple-Event Probabilities
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Each probability must lie between 0 and 1.Sum of probabilities for all simple events in sample space 𝑆 must equal 1.
Calculating Probability of an Event
Steps to Calculate Probability of an Event
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List all simple events in the sample space.Assign probabilities to each simple event.Identify simple events resulting in the event of interest.Sum the probabilities of these simple events.
Important Tips
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Include all simple events in the sample space.Assign realistic probabilities to simple events.Avoid omissions and ensure accuracy in probabilities.
Using Counting Rules
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When dealing with large sample spaces, use counting rules to identify and list all simple events accurately.
4.3. Useful Counting Rules
The mn Rule
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Definition: For two-stage experiments, with m outcomes in the first stage and n in the second, total outcomes = mn.
The Extended mn Rule
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Definition: For k-stage experiments with 𝑛1,𝑛2,...,𝑛𝑘 outcomes, total outcomes = 𝑛1×𝑛2×...×𝑛𝑘
Permutations
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Definition: Orderings of objects. Formula: 𝑃𝑟𝑛=𝑛!(𝑛−𝑟)!
Combinations
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Definition: Selections of objects without regard to order. Formula: 𝐶𝑟𝑛=𝑛!𝑟!(𝑛−𝑟)!Relation to Permutations: 𝐶𝑟𝑛=𝑃𝑟𝑛/𝑟!
Practical Tips
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Using Calculators: Instructions for TI-83/84 Plus to compute permutations, combinations, and factorials.Importance of Valid Counts: Ensure all simple events are included and correctly assigned probabilities.
