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 19.1% daily failure rate. Com (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? 0.191 (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? 0.03648 (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.)

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 19.1% daily failure rate. Com (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?
0.191 (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?
0.03648 (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.)

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 $19.1 \%$ daily failure rate. Com (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? 0.191 (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? 0.03648 (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.)

Solution

Solution Steps

Step 1: Probability of Alarm Clock Failure

The probability that the student's alarm clock will not work on the morning of an important final exam is given by the failure rate $ q $. Since the failure rate is $ 19.1\% $, we have:

\[ q = 0.191 \]

Thus, the probability that the alarm clock will not work is:

\[ \text{Probability (not working)} = q = 0.809 \]

Step 2: Probability of Two Alarm Clocks Failing

If the student has two independent alarm clocks, the probability that both will fail is calculated using the formula for independent events:

\[ P(\text{both fail}) = q^2 = (0.191)^2 = 0.03648 \]

Step 3: Probability of Three Alarm Clocks Failing

For three independent alarm clocks, the probability that all three will fail is given by:

\[ P(\text{all three fail}) = q^3 = (0.191)^3 = 0.00697 \]

Final Answer

Part (a): The probability that the student's alarm clock will not work is $ \boxed{0.809} $.
Part (b): The probability that both alarm clocks will fail is $ \boxed{0.03648} $.
Part (c): The probability that all three alarm clocks will fail is $ \boxed{0.00697} $.

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