Question

Redundancy. Exercises 25 and 26 involve redundancy.

Redundancy in Hospital Generators Hospitals typically require backup generators to provide electricity in the event of a power outage. Assume that emergency backup generators fail 22% of the times when they are needed (based on data from Arshad Mansoor, senior vice president with the Electric Power Research Institute). A hospital has two backup generators so that power is available if one of them fails during a power outage.

a. Find the probability that both generators fail during a power outage.

b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital?

Short answer

a. The probability that both generators fail during a power outage is equal to 0.0484.

b. The probability that a generator will work during a power outage is equal to 0.952.

c. The 95% chance that there will be a working generator during a power outage is not high enough. This implies a 5% chance of total failure, which can be considered a great risk in a hospital as it can lead to the loss of lives.

Step-by-step solution

Given information

The failure rate for a generator in a power outage at a hospital is 22%.

Multiplication rule of probability

For a set of two or more events, the probability that they will occur together can be calculated as follows:

$P\left(AandB\right)=P\left(A\right)\times P\left(B|A\right)$

Here, $P\left(B|A\right)$shows the probability of occurrence of B when A has already occurred.

Complementary events

For an event A, the event of occurrence of “not A” or “$\left(\overline{A}\right)$”is computed as follows:

$P\left(\overline{A}\right)=1-P\left(A\right)$

Here, A and $\left(\overline{A}\right)$can be considered as complementary events.

Compute the probability for the failure of both generators

a.

Let A be the event of failure of the first generator during a power outage.

The probability of failure of the first generator during a power outage is given by:

$$\begin{gathered} P\left(A\right)=\frac{22}{100} \\ =0.22 \end{gathered}$$

Let B be the event of failure of the second generator.

It is known that events A and B appear independently.

Therefore,$P\left(B\left|A\right.\right)=P\left(B\right)$ .

The probability of failure of the second generator, once the first generator fails during a power outage, is given by:

$$\begin{gathered} P\left(B\right)=\frac{22}{100} \\ =0.22 \end{gathered}$$

The probability of failure of both the generators is given by:

$$\begin{gathered} P\left(AandB\right)=P\left(A\right)\times P\left(B\right) \\ =0.22\times 0.22 \\ =0.0484 \end{gathered}$$

Therefore, the probability of failure of both the generators in a power outage is equal to 0.0484.

Compute the probability of having a working generator

b.

If A is the event of failure of a generator, then $\left(\overline{A}\right)$is the event that the generator works.

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-0.22 \\ =0.78 \end{gathered}$$

As there are a total of two generators, the probability of having a working generator in a power outage can be divided into three cases:

• Both the generators are working:

$$\begin{gathered} P\left(\overline{A}and\overline{B}\right)=P\left(\overline{A}\right)\times P\left(\overline{B}\right) \\ =0.78\times 0.78 \\ =0.6084 \end{gathered}$$

• The first one is working and the second one is not working:

$$\begin{gathered} P\left(\overline{A}\text{and}B\right)=P\left(\overline{A}\right)\times P\left(B\right) \\ =0.78\times 0.22 \\ =0.1716 \end{gathered}$$

• The first one is not working and the second one is working:

$$\begin{gathered} P\left(Aand\overline{B}\right)=P\left(A\right)\times P\left(\overline{B}\right) \\ =0.22\times 0.78 \\ =0.1716 \end{gathered}$$

Adding the above three probabilities, the probability of a working generator during a power outage is obtained.

$$\begin{gathered} P\left(workinggenerator\right)=0.6084+0.1716+0.1716 \\ =0.9516 \end{gathered}$$

Therefore, the probability of having a working generator in a power outage is equal to 0.952.

Interpretation of probability

There is a 5% probability of total failure, with two backup generators at the hospital. Thus, there is an approximately 5% chance that none of the two generators will work during a power outage. The event cannot be considered rare.

This can be considered a very high probability in a hospital as there is a greater risk of losing lives if there is a power outage.