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 18.5% 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?
0.185 (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?
0.03423 (round to five decimal places as needed.)
c. what is the probability of not being awakened if the student uses three independent alarm clocks?
0.00633 (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:

When there is redundancy (multiple alarm clocks), the probability of all failing (not being awakened) decreases. For part b, using two clocks, the probability of both failing is \(0.185\times0.185 = 0.03423\). For part c, with three clocks, it is \(0.185\times0.185\times0.185=0.00633\). A probability of \(0.00633\) (for three clocks) is much lower than \(0.185\) (for one clock). Total malfunction (all clocks failing) is not impossible (probability is non - zero), but it is unlikely.

Answer:

C. Yes, because total malfunction would not be impossible, but it would be unlikely.