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 11.4% 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? (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?
a. yes, because you can always be certain that at least one alarm clock will work.
b. yes, because total malfunction would not be impossible, but it would be unlikely.
c. no, because the malfunction of both is equally or more likely than the malfunction of one.
d. no, because total malfunction would still not be unlikely.

Explanation:

Step1: Identify single - clock failure rate

The daily failure rate of a single alarm clock is given as 11.4% or 0.114.

Step2: Calculate probability for part a

The probability that a single alarm clock will not work is the given failure rate. So for part a, the probability $P_1 = 0.114$.

Step3: Calculate probability for part b

Since the two alarm - clocks are independent events, the probability that both fail is the product of their individual failure probabilities. So $P_2=0.114\times0.114 = 0.012996\approx0.01300$.

Step4: Calculate probability for part c

For three independent alarm - clocks, the probability that all three fail is the product of their individual failure probabilities. So $P_3 = 0.114\times0.114\times0.114=0.001480344\approx0.00148$.

Step5: Analyze part d

The probability of a single - clock failure is 0.114, the probability of two - clock failure is 0.01300, and the probability of three - clock failure is 0.00148. Total malfunction is less likely with more alarm clocks.

Answer:

a. 0.114
b. 0.01300
c. 0.00148
d. B. Yes, because total malfunction would not be impossible, but it would be unlikely.