Question 14
Question 15
Question 16
Question 17
Question 18
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 $17.1 \%$ 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? 0.171 (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?
0.02924 (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.)

Step 1 ：The problem is based on the principle of redundancy, which is used to improve system reliability through redundant or backup components. In this case, the system is a student's alarm clock, which has a daily failure rate of 17.1%.

Step 2 ：The probability that the student's alarm clock will not work on the morning of an important final exam is 0.171.

Step 3 ：If the student has two such alarm clocks, the probability that they both fail on the morning of an important final exam is 0.02924.

Step 4 ：We are asked to find the probability of not being awakened if the student uses three independent alarm clocks. This would occur if all three alarm clocks fail. Since the alarm clocks are independent, the probability of all three failing is the product of their individual failure probabilities.

Step 5 ：Let's denote the failure rate as \(p\), which is 0.171. The probability of all three alarm clocks failing is \(p^3\).

Step 6 ：Substituting the given value into the formula, we get \(0.171^3 = 0.005000211000000001\).

Step 7 ：Rounding to five decimal places as needed, the final answer is \(\boxed{0.00500}\).