Question

the principle of redundancy is used when system reliability is improved through redundant or backup components. assume that a students alarm clock has a 10.1% daily failure rate. complete parts (a) through (d) below.
a. what is the probability that the students alarm clock will not work on the morning of an important final exam? 0.101 (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? 0.01020 (round to five decimal places as needed.)
c. what is the probability of not being awakened if the student uses three independent alarm clocks? 0.00103 (round to five decimal places as needed.)
d. do the second and third alarm clocks result in greatly improved reliability?
a. no, because total malfunction would still not be unlikely.
b. yes, because you can always be certain that at least one alarm clock will work.
c. no, because the malfunction of both is equally or more likely than the malfunction of one.
d. yes, because total malfunction would not be impossible, but it would be unlikely.

Explanation:

Step1: Identify failure rate

The daily failure rate of one alarm - clock is $p = 0.181$.

Step2: Calculate probability of two - clock failure

Since the two alarms are independent, the probability that both fail is $P(\text{both fail})=p\times p$. Substituting $p = 0.181$, we get $P(\text{both fail})=0.181\times0.181 = 0.032761\approx0.03276$.

Step3: Calculate probability of three - clock failure

Since the three alarms are independent, the probability that all three fail is $P(\text{all three fail})=p\times p\times p$. Substituting $p = 0.181$, we get $P(\text{all three fail})=0.181\times0.181\times0.181=0.005931941\approx0.00593$.

Step4: Analyze reliability improvement

The probability of one - clock failure is $0.181$, the probability of two - clock failure is $0.03276$, and the probability of three - clock failure is $0.00593$. Total malfunction is less likely with more alarms. So, the second and third alarm clocks result in greatly improved reliability because total malfunction would not be impossible, but it would be unlikely.

Answer:

a. $0.181$
b. $0.03276$
c. $0.00593$
d. D. Yes, because total malfunction would not be impossible, but it would be unlikely.