Page 445 #6b An experiment in which the three mutually exclusive events A, B, and C form a partition of the uniform sample space S is depicted in the diagram below. (Round your answers to three decimal places.

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...
icon
Related questions
icon
Concept explainers
Question

Page 445 #6b

An experiment in which the three mutually exclusive events AB, and C form a partition of the uniform sample space S is depicted in the diagram below. (Round your answers to three decimal places.)

**Page 445 #6b**

An experiment in which the three mutually exclusive events \( A, B, \) and \( C \) form a partition of the uniform sample space \( S \) is depicted in the diagram below. (Round your answers to three decimal places.)

![Diagram](attachment_name)

The diagram is divided as follows:
- A rectangular space labeled \( S \) contains three vertical sections labeled \( A, B, \) and \( C \), each containing numerical values.
- Inside the rectangle, there is an oval labeled \( D \) with its own numerical values.

The numerical values are:
- In section \( A \) (outside the oval \( D \)): 25
- In section \( B \) (overlapping with \( D \)): 20 inside \( D \) and 20 outside \( D \)
- In section \( C \) (overlapping with \( D \)): 15 inside \( D \) and 20 outside \( D \) 

(b) **Find \( P(B \mid D^c) = \)**

**Explanation:**
- \( D^c \) refers to the complement of the event \( D \), which includes all outcomes not in \( D \). 
- The values relevant to \( D^c \) are those outside the oval, specifically: 25 from \( A \), 20 from \( B \), and 20 from \( C \).
- To calculate \( P(B \mid D^c) \), use the formula:
\[ 
P(B \mid D^c) = \frac{P(B \cap D^c)}{P(D^c)} 
\]
- Calculate \( P(B \cap D^c) \), which is the probability of event \( B \) occurring outside \( D \): 20.
- Calculate \( P(D^c) \), the total of all probabilities outside \( D \): 25 (from \( A \)) + 20 (from \( B \)) + 20 (from \( C \)) = 65.

Finally, substituting these into the formula:
\[ 
P(B \mid D^c) = \frac{20}{65} 
\]
- Simplify and round the calculation to three decimal places.
Transcribed Image Text:**Page 445 #6b** An experiment in which the three mutually exclusive events \( A, B, \) and \( C \) form a partition of the uniform sample space \( S \) is depicted in the diagram below. (Round your answers to three decimal places.) ![Diagram](attachment_name) The diagram is divided as follows: - A rectangular space labeled \( S \) contains three vertical sections labeled \( A, B, \) and \( C \), each containing numerical values. - Inside the rectangle, there is an oval labeled \( D \) with its own numerical values. The numerical values are: - In section \( A \) (outside the oval \( D \)): 25 - In section \( B \) (overlapping with \( D \)): 20 inside \( D \) and 20 outside \( D \) - In section \( C \) (overlapping with \( D \)): 15 inside \( D \) and 20 outside \( D \) (b) **Find \( P(B \mid D^c) = \)** **Explanation:** - \( D^c \) refers to the complement of the event \( D \), which includes all outcomes not in \( D \). - The values relevant to \( D^c \) are those outside the oval, specifically: 25 from \( A \), 20 from \( B \), and 20 from \( C \). - To calculate \( P(B \mid D^c) \), use the formula: \[ P(B \mid D^c) = \frac{P(B \cap D^c)}{P(D^c)} \] - Calculate \( P(B \cap D^c) \), which is the probability of event \( B \) occurring outside \( D \): 20. - Calculate \( P(D^c) \), the total of all probabilities outside \( D \): 25 (from \( A \)) + 20 (from \( B \)) + 20 (from \( C \)) = 65. Finally, substituting these into the formula: \[ P(B \mid D^c) = \frac{20}{65} \] - Simplify and round the calculation to three decimal places.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Conditional Probability, Decision Trees, and Bayes' Theorem
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability
A First Course in Probability
Probability
ISBN:
9780321794772
Author:
Sheldon Ross
Publisher:
PEARSON