A First Course in Probability (10th Edition)
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ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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![## Understanding Redundancy and Probabilities
### Problem Statement
**5)** The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that your alarm clock has a 0.9 probability of working on any given morning.
**a)** What is the probability that your alarm clock will *not* work on the morning of an important final exam?
### Explanation
In this problem, we are examining the reliability of a system (in this case, an alarm clock) and the concept of redundancy. The problem gives a probability of 0.9 for the alarm clock working, from which we need to determine the probability of it not working on a specific morning, such as the morning of a final exam.
### Solution Approach
To find the probability that the alarm clock will not work, we can use the concept of complementary probability:
- Probability of the alarm clock working: \( P(\text{Working}) = 0.9 \)
- Probability of the alarm clock not working: \( P(\text{Not Working}) = 1 - P(\text{Working}) \)
Therefore:
\[ P(\text{Not Working}) = 1 - 0.9 = 0.1 \]
So, there is a 0.1 probability that your alarm clock will not function on the morning of an important final exam.](https://content.bartleby.com/qna-images/question/5018b380-448e-4214-adf4-74ec1b10342e/a3da86ea-9bef-4c0c-841e-2128e86f487f/lqc7ap_processed.jpeg)
Transcribed Image Text:## Understanding Redundancy and Probabilities
### Problem Statement
**5)** The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that your alarm clock has a 0.9 probability of working on any given morning.
**a)** What is the probability that your alarm clock will *not* work on the morning of an important final exam?
### Explanation
In this problem, we are examining the reliability of a system (in this case, an alarm clock) and the concept of redundancy. The problem gives a probability of 0.9 for the alarm clock working, from which we need to determine the probability of it not working on a specific morning, such as the morning of a final exam.
### Solution Approach
To find the probability that the alarm clock will not work, we can use the concept of complementary probability:
- Probability of the alarm clock working: \( P(\text{Working}) = 0.9 \)
- Probability of the alarm clock not working: \( P(\text{Not Working}) = 1 - P(\text{Working}) \)
Therefore:
\[ P(\text{Not Working}) = 1 - 0.9 = 0.1 \]
So, there is a 0.1 probability that your alarm clock will not function on the morning of an important final exam.
Transcribed Image Text:b) If you have two such alarm clocks, what is the probability that they both fail on the morning of the important exam?
c) With one alarm clock, you have a 0.9 probability of being awakened. What is the probability of being awakened if you use two alarm clocks?
d) Does a second alarm clock result in greatly improved reliability?
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