The sample space S is depicted by the Venn Diagram below. P(A) = 0.35, P(B) = 0.67,P(C) = 0.13 P(An B) = 0.05, P (A n C) = 0.13, P (B n C) = 0.07 Given that P(A UBUC) = 1, calculate the following probabilities: (a) P (AUB) (b) P(BIC) (c) P(An BNC) A с B
The sample space S is depicted by the Venn Diagram below. P(A) = 0.35, P(B) = 0.67,P(C) = 0.13 P(An B) = 0.05, P (A n C) = 0.13, P (B n C) = 0.07 Given that P(A UBUC) = 1, calculate the following probabilities: (a) P (AUB) (b) P(BIC) (c) P(An BNC) A с B
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...
Related questions
Question
![### Understanding Venn Diagrams and Probabilities in Set Theory
The sample space \( S \) is depicted by the Venn Diagram below.
Given probabilities:
\[ P(A) = 0.35, \; P(B) = 0.67, \; P(C) = 0.13 \]
\[ P(A \cap B) = 0.05, \; P(A \cap C') = 0.13, \; P(B \cap C) = 0.07, \; P(A \cup B \cup C) = 1 \]
The goal is to calculate the following probabilities:
(a) \( P(A \cup B) \)
(b) \( P(B|C) \)
(c) \( P(A \cap B \cap C) \)
#### Venn Diagram Explanation
The Venn diagram included illustrates three events \( A \), \( B \), and \( C \), each represented by a circle. The intersections of these circles depict the interactions between events, allowing us to visualize and calculate combined probabilities.
**(a) Calculating \( P(A \cup B) \):**
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Using the given values:
\[ P(A \cup B) = 0.35 + 0.67 - 0.05 \]
\[ P(A \cup B) = 0.97 \]
**(b) Calculating \( P(B|C) \):**
\[ P(B|C) = \frac{P(B \cap C)}{P(C)} \]
Using the given values:
\[ P(B|C) = \frac{0.07}{0.13} \]
\[ P(B|C) = 0.538 \] (approx.)
**(c) Calculating \( P(A \cap B \cap C) \):**
From the given data:
\[
P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C') + P(A \cap B \cap C)
\]
Given \( P(A \cup B \cup C) = 1 \):
\[ 1 = 0.35 + 0.67 + 0.13 - 0.05 -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9de804fd-1a97-452a-9cf6-b7d2ceae8a21%2F3b0a6265-7b0e-478a-8429-b28f95cd1e15%2Fv3qv47h_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Venn Diagrams and Probabilities in Set Theory
The sample space \( S \) is depicted by the Venn Diagram below.
Given probabilities:
\[ P(A) = 0.35, \; P(B) = 0.67, \; P(C) = 0.13 \]
\[ P(A \cap B) = 0.05, \; P(A \cap C') = 0.13, \; P(B \cap C) = 0.07, \; P(A \cup B \cup C) = 1 \]
The goal is to calculate the following probabilities:
(a) \( P(A \cup B) \)
(b) \( P(B|C) \)
(c) \( P(A \cap B \cap C) \)
#### Venn Diagram Explanation
The Venn diagram included illustrates three events \( A \), \( B \), and \( C \), each represented by a circle. The intersections of these circles depict the interactions between events, allowing us to visualize and calculate combined probabilities.
**(a) Calculating \( P(A \cup B) \):**
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Using the given values:
\[ P(A \cup B) = 0.35 + 0.67 - 0.05 \]
\[ P(A \cup B) = 0.97 \]
**(b) Calculating \( P(B|C) \):**
\[ P(B|C) = \frac{P(B \cap C)}{P(C)} \]
Using the given values:
\[ P(B|C) = \frac{0.07}{0.13} \]
\[ P(B|C) = 0.538 \] (approx.)
**(c) Calculating \( P(A \cap B \cap C) \):**
From the given data:
\[
P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C') + P(A \cap B \cap C)
\]
Given \( P(A \cup B \cup C) = 1 \):
\[ 1 = 0.35 + 0.67 + 0.13 - 0.05 -
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON