Redundancy Using Braun battery-powered alarm clocks, the author estimates that the probability of failure on any given day is . (a) What is the probability that the alarm clock works for an important event? (b) When he uses two alarm clocks for important events, what is the probability that at least one of them works?

The probability of failure of an alarm clock on any given day is $\frac{1}{1000}$ .

The concept of probability identifies the measurement of chances for the occurrence of a specific event. The value would never be lower than zero or greater than one.

Consequently, the probability of occurrence for the complementary event is the probability of the specific event subtracted from one.

If multiple events occur jointly, the product of the probabilities of each of the events constitutes for the probability of the joint occurrence of all events. The idea is stated as the multiplication rule.

Define event A as the event that the alarm clock fails on any given day.

The complementary event is , which implies the event that the alarm clock does not fail on a specific day.

Thus,

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-\frac{1}{1000} \\ =\frac{999}{1000} \\ =0.999 \end{gathered}$$

Thus, the probability that an alarm clock works on an important event is 0.999.

Let E be the event that none of the alarm clocks works.

The complementary event would be defined as $\overline{E}$, which states that at least one alarm clock works.

Thus, the probability that at least one alarm clock works is one minus the probability none of the alarm clocks works.

Also, as both alarm clocks are independent, the probability that none of the alarm clocks works is the product of the probability of failure of two alarm clocks.

Thus, the probability that none of the alarm clocks works is:

$$\begin{gathered} P\left(E\right)=P\left(A\right)\times P\left(A\right) \\ =\frac{1}{1000}\times \frac{1}{1000} \\ =\frac{1}{1000000} \end{gathered}$$

Consequently, the probability that at least one alarm clock works is:

$$\begin{gathered} P\left(\overline{E}\right)=1-P\left(E\right) \\ =1-\frac{1}{1000000} \\ =\frac{999999}{1000000} \\ =0.999999 \end{gathered}$$

Thus, the probability that at least one alarm clock works is 0.999999.