V Corporation produces Blu-ray players that have three major components: optical system, disc drive mechanism, and internal electronics. All the components should work for the player to function. The optical system has a reliability of 95%, the disc drive mechanism has a reliability of 90%, and the internal electronics has a reliability of 70%. The internal electronics has a backup component with the same reliability of 70%. What is the reliability of V Corporation's Blu-ray players?
V Corporation produces Blu-ray players that have three major components: optical system, disc drive mechanism, and internal electronics. All the components should work for the player to function. The optical system has a reliability of 95%, the disc drive mechanism has a reliability of 90%, and the internal electronics has a reliability of 70%. The internal electronics has a backup component with the same reliability of 70%. What is the reliability of V Corporation's Blu-ray players?
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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![**Problem Statement:**
V Corporation produces Blu-ray players that have three major components: optical system, disc drive mechanism, and internal electronics. All the components should work for the player to function.
- The optical system has a reliability of 95%.
- The disc drive mechanism has a reliability of 90%.
- The internal electronics has a reliability of 70%.
The internal electronics has a backup component with the same reliability of 70%.
**Question:** What is the reliability of V Corporation's Blu-ray players?
**Options:**
- 0.778
- 0.855
- 0.599
- 0.550
**Explanation:**
To find the overall reliability of the Blu-ray player, we need to determine the reliability of a system where all parts must function correctly.
1. **Calculate the reliability of the internal electronics with a backup:**
The reliability \( R \) for a component with a backup can be calculated using:
\[
R = 1 - (1 - R_1) \times (1 - R_2)
\]
Where \( R_1 \) and \( R_2 \) are the reliabilities of the primary and backup components, respectively.
For the internal electronics:
\[
R = 1 - (1 - 0.7) \times (1 - 0.7)
\]
\[
R = 1 - 0.3 \times 0.3
\]
\[
R = 1 - 0.09 = 0.91
\]
2. **Calculate the overall system reliability:**
Since all components must work:
\[
R_{\text{system}} = R_{\text{optical}} \times R_{\text{disc}} \times R_{\text{internal}}
\]
Given:
- \( R_{\text{optical}} = 0.95 \)
- \( R_{\text{disc}} = 0.90 \)
- \( R_{\text{internal}} = 0.91 \)
Therefore:
\[
R_{\text{system}} = 0.95 \times 0.90 \times 0.91
\]
3. **Calculate the result:**
\[
R_{\text{system}} = 0.778
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Transcribed Image Text:**Problem Statement:**
V Corporation produces Blu-ray players that have three major components: optical system, disc drive mechanism, and internal electronics. All the components should work for the player to function.
- The optical system has a reliability of 95%.
- The disc drive mechanism has a reliability of 90%.
- The internal electronics has a reliability of 70%.
The internal electronics has a backup component with the same reliability of 70%.
**Question:** What is the reliability of V Corporation's Blu-ray players?
**Options:**
- 0.778
- 0.855
- 0.599
- 0.550
**Explanation:**
To find the overall reliability of the Blu-ray player, we need to determine the reliability of a system where all parts must function correctly.
1. **Calculate the reliability of the internal electronics with a backup:**
The reliability \( R \) for a component with a backup can be calculated using:
\[
R = 1 - (1 - R_1) \times (1 - R_2)
\]
Where \( R_1 \) and \( R_2 \) are the reliabilities of the primary and backup components, respectively.
For the internal electronics:
\[
R = 1 - (1 - 0.7) \times (1 - 0.7)
\]
\[
R = 1 - 0.3 \times 0.3
\]
\[
R = 1 - 0.09 = 0.91
\]
2. **Calculate the overall system reliability:**
Since all components must work:
\[
R_{\text{system}} = R_{\text{optical}} \times R_{\text{disc}} \times R_{\text{internal}}
\]
Given:
- \( R_{\text{optical}} = 0.95 \)
- \( R_{\text{disc}} = 0.90 \)
- \( R_{\text{internal}} = 0.91 \)
Therefore:
\[
R_{\text{system}} = 0.95 \times 0.90 \times 0.91
\]
3. **Calculate the result:**
\[
R_{\text{system}} = 0.778
\]
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