If P(Aº) =0.4 P(B | A) =0.30 and P(B© | A°) =0.1 P(A U Bº) is .74 .64 .56 .48
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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![### Probability Problem for Educational Website
**Problem Statement:**
Given the following probabilities:
- \( P(A^c) = 0.4 \)
- \( P(B \mid A) = 0.30 \)
- \( P(B^c \mid A^c) = 0.1 \)
Find the value of \( P(A \cup B^c) \).
**Options:**
- A) 0.74
- B) 0.64
- C) 0.56
- D) 0.48
**Explanation of Probabilities and Terms:**
1. \( P(A^c) \) denotes the probability that event \( A \) does not occur.
2. \( P(B \mid A) \) denotes the conditional probability that event \( B \) occurs given that event \( A \) has occurred.
3. \( P(B^c \mid A^c) \) denotes the conditional probability that event \( B \) does not occur given that event \( A \) has not occurred.
4. \( P(A \cup B^c) \) represents the probability that either event \( A \) occurs or event \( B \) does not occur (or both).
**Solution Approach:**
To calculate \( P(A \cup B^c) \), we use the following formula:
\[ P(A \cup B^c) = 1 - P(A^c \cap B) \]
Firstly, determine the probability \( P(A^c \cap B) \):
\[ P(A^c \cap B) = P(B \mid A^c) \cdot P(A^c) \]
However, we have \( P(B^c \mid A^c) \), thus:
\[ P(B \mid A^c) = 1 - P(B^c \mid A^c) = 1 - 0.1 = 0.9 \]
Therefore,
\[ P(A^c \cap B) = 0.9 \cdot 0.4 = 0.36 \]
Now, calculate \( P(A \cup B^c) \):
\[ P(A \cup B^c) = 1 - P(A^c \cap B) = 1 - 0.36 = 0.64 \]
**Correct Answer:**
- B) 0.64
This detailed](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd529b6da-11b0-40e5-85fb-f211e59fb34e%2F2dd0f770-243a-4009-974d-20a6851c293f%2Fzegr8k3.png&w=3840&q=75)
Transcribed Image Text:### Probability Problem for Educational Website
**Problem Statement:**
Given the following probabilities:
- \( P(A^c) = 0.4 \)
- \( P(B \mid A) = 0.30 \)
- \( P(B^c \mid A^c) = 0.1 \)
Find the value of \( P(A \cup B^c) \).
**Options:**
- A) 0.74
- B) 0.64
- C) 0.56
- D) 0.48
**Explanation of Probabilities and Terms:**
1. \( P(A^c) \) denotes the probability that event \( A \) does not occur.
2. \( P(B \mid A) \) denotes the conditional probability that event \( B \) occurs given that event \( A \) has occurred.
3. \( P(B^c \mid A^c) \) denotes the conditional probability that event \( B \) does not occur given that event \( A \) has not occurred.
4. \( P(A \cup B^c) \) represents the probability that either event \( A \) occurs or event \( B \) does not occur (or both).
**Solution Approach:**
To calculate \( P(A \cup B^c) \), we use the following formula:
\[ P(A \cup B^c) = 1 - P(A^c \cap B) \]
Firstly, determine the probability \( P(A^c \cap B) \):
\[ P(A^c \cap B) = P(B \mid A^c) \cdot P(A^c) \]
However, we have \( P(B^c \mid A^c) \), thus:
\[ P(B \mid A^c) = 1 - P(B^c \mid A^c) = 1 - 0.1 = 0.9 \]
Therefore,
\[ P(A^c \cap B) = 0.9 \cdot 0.4 = 0.36 \]
Now, calculate \( P(A \cup B^c) \):
\[ P(A \cup B^c) = 1 - P(A^c \cap B) = 1 - 0.36 = 0.64 \]
**Correct Answer:**
- B) 0.64
This detailed
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