Question

Redundancy in Stadium Generators Large stadiums rely on backup generators to provide electricity in the event of a power failure. 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 stadium has three backup generators so that power is available if at least one of them works in a power failure. Find the probability of having at least one of the backup generators working given that a power failure has occurred. Does the result appear to be adequate for the stadium’s needs?

Short answer

The probability that at least one will work out of three generators during a power failure is equal to 0.989.

As the probability value is high, it seems sufficient for the stadium’s needs.

Step-by-step solution

Given information

The emergency backup generators fail 22% of the time whenever needed.

The number of backup generators with the stadium is 3.

Define the event of “at least one”

An event of at least one occurrence implies that a specific situation occurs once or more than once.The probability of the event is calculated by computing the probability that the event does not occur and then subtracting it from 1.

Mathematically, if A is an event,

$P\left(Aoccurringatleastonce\right)=1-P\left(Anotoccurring\right)$

Compute the probability that at least one backup generator works

Let A be the event that a backup generator fails to work.

It has the following probability:

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

The probability that among the three generators that all will fail.

$$\begin{gathered} P\left(\text{all}\text{will}\text{fail}\right)=P\left(A\right)\times P\left(A\right)\times P\left(A\right) \\ =0.22\times 0.22\times 0.22 \\ =0.0106 \end{gathered}$$

The probability that out of three generators, at least one will work during a power failure is computed as follows:

$$\begin{gathered} P\left(\text{at}\text{least}\text{one}\text{will}\text{work}\right)=1-P\left(\text{all}\text{will}\text{fail}\right) \\ =1-0.0106 \\ =0.9894 \end{gathered}$$

Therefore, the probability that at least one will work in a power failure out of the three backup generators is 0.989.

Interpret the probability

The probability that at least one of the three backup generators works during a power failure is relatively high, 0.989 (close to 1).

The probability value seems adequate to fulfill the needs of the stadium during a power failure as it is sufficiently high.