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? (round to five decimal places as needed.)

Explanation:

Step1: Identify the probability of one alarm clock failing

Given that the daily failure rate of one alarm clock is \(p = 0.185\).

Step2: Use the multiplication rule for independent events

If two events \(A\) and \(B\) are independent, the probability that both \(A\) and \(B\) occur is \(P(A\cap B)=P(A)\times P(B)\). Here, let \(A\) be the event that the first alarm clock fails and \(B\) be the event that the second alarm clock fails. Since the failures of the two alarm - clocks are independent events, and \(P(A) = P(B)=0.185\).
The probability that both fail is \(P = 0.185\times0.185\).

$$P=(0.185)^2=0.034225\approx0.03423$$

Answer:

\(0.03423\)