Questions: The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that a student's alarm clock has a 11.7 % daily failure rate. Complete parts (a) through (d) below. a. What is the probability that the student's alarm clock will not work on the morning of an important final exam? (Round to three decimal places as needed.) b. If the student has two such alarm clocks, what is the probability that they both fail on the morning of an important final exam? (Round to five decimal places as needed.) c. What is the probability of not being awakened if the student uses three independent alarm clocks? (Round to five decimal places as needed.) d. Do the second and third alarm clocks result in greatly improved reliability?

The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that a student's alarm clock has a 11.7 % daily failure rate. Complete parts (a) through (d) below.
a. What is the probability that the student's alarm clock will not work on the morning of an important final exam? (Round to three decimal places as needed.)
b. If the student has two such alarm clocks, what is the probability that they both fail on the morning of an important final exam?
(Round to five decimal places as needed.)
c. What is the probability of not being awakened if the student uses three independent alarm clocks?
(Round to five decimal places as needed.)
d. Do the second and third alarm clocks result in greatly improved reliability?

Transcript text: The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that a student's alarm clock has a $11.7 \%$ daily failure rate. Complete parts (a) through (d) below. a. What is the probability that the student's alarm clock will not work on the morning of an important final exam? $\square$ (Round to three decimal places as needed.) b. If the student has two such alarm clocks, what is the probability that they both fail on the morning of an important final exam? $\square$ (Round to five decimal places as needed.) c. What is the probability of not being awakened if the student uses three independent alarm clocks? $\square$ (Round to five decimal places as needed.) d. Do the second and third alarm clocks result in greatly improved reliability?

Solution

Solution Steps

Step 1: Calculate Single Component Failure Probability

The probability that a single component will fail is directly given by its failure rate $f$. Given $f = 0.117$, the probability of failure is $P(Failure) = f = 0.117$.

Final Answer: The probability for the event 'single_failure' is 0.117.

Step 1: Calculate All Components Failure Probability

For $n = 2$ independent components each with a failure rate of $f = 0.117$, the probability that all $n$ components fail is the product of their individual failure probabilities. Thus, $P(All\ Fail) = f^n = 0.117^2 = 0.0137$.

Final Answer: The probability for the event 'all_fail' is 0.0137.

Step 1: Calculate At Least One Component Not Failing Probability

The probability that at least one of $n = 3$ independent components does not fail is complementary to the probability that all components fail. Thus, $P(At\ Least\ One\ Success) = 1 - P(All\ Fail) = 1 - 0.117^3 = 0.998$.

Final Answer: The probability for the event 'at_least_one_success' is 0.998.

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