Determine whether the events A and B are independent. P(A) = 0.2, P(B) = 0.4, P(A n B) = 0.48 Yes Ο NO
Determine whether the events A and B are independent. P(A) = 0.2, P(B) = 0.4, P(A n B) = 0.48 Yes Ο NO
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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determine the following probability question
![**Determine whether the events A and B are independent.**
Given:
- \( P(A^c) = 0.2 \)
- \( P(B^c) = 0.4 \)
- \( P(A \cap B) = 0.48 \)
Options:
- ( ) Yes
- ( ) No
**Explanation:**
To determine if events \( A \) and \( B \) are independent, we need to check if the following condition holds:
\[ P(A \cap B) = P(A) \cdot P(B) \]
First, calculate \( P(A) \) and \( P(B) \):
- \( P(A) = 1 - P(A^c) = 1 - 0.2 = 0.8 \)
- \( P(B) = 1 - P(B^c) = 1 - 0.4 = 0.6 \)
Now, calculate \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = 0.8 \cdot 0.6 = 0.48 \]
Since \( P(A \cap B) = 0.48 \), which equals \( P(A) \cdot P(B) \), the events \( A \) and \( B \) are independent.
Correct answer:
- ( X ) Yes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd69e9d70-82b1-4b79-b09d-01780faac5b4%2F585131af-8c7d-4572-9ebb-b412313cbbe0%2Frzdt5mu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine whether the events A and B are independent.**
Given:
- \( P(A^c) = 0.2 \)
- \( P(B^c) = 0.4 \)
- \( P(A \cap B) = 0.48 \)
Options:
- ( ) Yes
- ( ) No
**Explanation:**
To determine if events \( A \) and \( B \) are independent, we need to check if the following condition holds:
\[ P(A \cap B) = P(A) \cdot P(B) \]
First, calculate \( P(A) \) and \( P(B) \):
- \( P(A) = 1 - P(A^c) = 1 - 0.2 = 0.8 \)
- \( P(B) = 1 - P(B^c) = 1 - 0.4 = 0.6 \)
Now, calculate \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = 0.8 \cdot 0.6 = 0.48 \]
Since \( P(A \cap B) = 0.48 \), which equals \( P(A) \cdot P(B) \), the events \( A \) and \( B \) are independent.
Correct answer:
- ( X ) Yes
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