B are defined on a common sample D30.56, find P(A and B).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Problem Statement:**
14. Events \( A \) and \( B \) are defined on a common sample space. If \( P(A) = 0.20 \), \( P(B) = 0.40 \), and \( P(A \text{ or } B) = 0.56 \), find \( P(A \text{ and } B) \).
**Explanation:**
To solve this problem, we need to use the formula for the probability of the union of two events:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
Given:
- \( P(A) = 0.20 \)
- \( P(B) = 0.40 \)
- \( P(A \text{ or } B) = 0.56 \)
Plugging into the formula:
\[ 0.56 = 0.20 + 0.40 - P(A \text{ and } B) \]
Solve for \( P(A \text{ and } B) \):
\[ 0.56 = 0.60 - P(A \text{ and } B) \]
\[ P(A \text{ and } B) = 0.60 - 0.56 \]
\[ P(A \text{ and } B) = 0.04 \]
Therefore, the probability that both events \( A \) and \( B \) occur is \( P(A \text{ and } B) = 0.04 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38cbb8d5-b2e4-4062-be8a-b15364eb39dc%2F89e22a51-a2b1-4a5b-b36a-67ecf8609ed1%2F34rss_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
14. Events \( A \) and \( B \) are defined on a common sample space. If \( P(A) = 0.20 \), \( P(B) = 0.40 \), and \( P(A \text{ or } B) = 0.56 \), find \( P(A \text{ and } B) \).
**Explanation:**
To solve this problem, we need to use the formula for the probability of the union of two events:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
Given:
- \( P(A) = 0.20 \)
- \( P(B) = 0.40 \)
- \( P(A \text{ or } B) = 0.56 \)
Plugging into the formula:
\[ 0.56 = 0.20 + 0.40 - P(A \text{ and } B) \]
Solve for \( P(A \text{ and } B) \):
\[ 0.56 = 0.60 - P(A \text{ and } B) \]
\[ P(A \text{ and } B) = 0.60 - 0.56 \]
\[ P(A \text{ and } B) = 0.04 \]
Therefore, the probability that both events \( A \) and \( B \) occur is \( P(A \text{ and } B) = 0.04 \).
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