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 18.1% 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?
0.181 (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?
0.03276 (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.)

Explanation:

Step1: Identify failure rate

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

Step2: Use multiplication rule for independent events

For three independent alarm - clocks, the probability that all three fail is given by the multiplication rule for independent events. If $A$, $B$, and $C$ are independent events, $P(A\cap B\cap C)=P(A)\times P(B)\times P(C)$. Here, the probability of each alarm - clock failing is $0.181$. So the probability that all three fail is $P = 0.181\times0.181\times0.181$.

Step3: Calculate the result

$P=(0.181)^3=0.181\times0.181\times0.181 = 0.00593$

Answer:

$0.00593$