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Let E, F, G be independent events. Prove that E is independent with each of the following sets: FUG, FUG, FnG, Fen Ge

Let E, F, G be independent events. Prove that E is independent with each of the following sets: FUG, FUG, FnG, Fen Ge

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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### Independence of Events and Set Operations Let \( E, F, G \) be independent events. Prove that \( E \) is independent with each of the following sets: 1. \( F \cup G^c \) 2. \( F^c \cup G^c \) 3. \( F \cap G^c \) 4. \( F^c \cap G^c \) #### Explanation: To prove that \( E \) is independent with these sets, recall the definition of independent events. Two events \( A \) and \( B \) are independent if and only if: \[ P(A \cap B) = P(A) \cdot P(B) \] For each of the given expressions, you will need to confirm that: \[ P(E \cap (Expression)) = P(E) \cdot P(Expression) \] where \( Expression \) refers to each of the sets provided. This involves using properties of probability, such as the complement rule \( P(A^c) = 1 - P(A) \), and the distributions of unions and intersections over probabilities.
Transcribed Image Text:### Independence of Events and Set Operations Let \( E, F, G \) be independent events. Prove that \( E \) is independent with each of the following sets: 1. \( F \cup G^c \) 2. \( F^c \cup G^c \) 3. \( F \cap G^c \) 4. \( F^c \cap G^c \) #### Explanation: To prove that \( E \) is independent with these sets, recall the definition of independent events. Two events \( A \) and \( B \) are independent if and only if: \[ P(A \cap B) = P(A) \cdot P(B) \] For each of the given expressions, you will need to confirm that: \[ P(E \cap (Expression)) = P(E) \cdot P(Expression) \] where \( Expression \) refers to each of the sets provided. This involves using properties of probability, such as the complement rule \( P(A^c) = 1 - P(A) \), and the distributions of unions and intersections over probabilities.
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