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 15.7% 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?
.157 (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?
.02465 (round to five decimal places as needed.)
c. what is the probability of not being awakened if the student uses three independent alarm clocks?
.00387 (round to five decimal places as needed.)
d. do the second and third alarm clocks result in greatly improved reliability?
a. no, because the malfunction of both is equally or more likely than the malfunction of one.
b. no, because total malfunction would still not be unlikely.
c. yes, because total malfunction would not be impossible, but it would be unlikely.
d. yes, because you can always be certain that at least one alarm clock will work.

Explanation:

Step1: Identify failure rate

The daily failure rate of one alarm clock is given as 15.7% or 0.157.

Step2: Calculate probability of two - clock failure

Since the failures of the two alarm clocks are independent events, the probability that both fail is the product of their individual failure probabilities. So $P(\text{both fail})=0.157\times0.157 = 0.024649\approx0.02465$.

Step3: Calculate probability of three - clock failure

For three independent alarm clocks, the probability that all three fail is $P(\text{all three fail})=0.157\times0.157\times0.157= 0.003869833\approx0.00387$.

Step4: Analyze reliability improvement

The probability of one - clock failure is 0.157. The probability of two - clock failure is 0.02465 and of three - clock failure is 0.00387. As the number of alarm clocks increases, the probability of total failure decreases significantly. Total malfunction is not impossible but is unlikely.

Answer:

a. 0.157
b. 0.02465
c. 0.00387
d. C. Yes, because total malfunction would not be impossible, but it would be unlikely.