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Chapter 4: Problem 22

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

Expert verified
The probability is approximately 98.94%. This seems adequate.

Step by step solution

01

- Define the problem

We need to find the probability that at least one of three backup generators works during a power failure, given that each generator has a 22% chance of failing when needed.
02

- Calculate the failure probability for one generator

The probability that one generator fails is given as 22%, so the probability that one generator works is \( P(W) = 1 - 0.22 = 0.78 \).
03

- Calculate the probability of all generators failing

We need to find the probability that all three generators fail. The probability of all three failing is \( P(F) = (0.22)^3 \).
04

- Use the multiplication rule

The probability that all three generators fail is \( (0.22)^3 = 0.010648 \).
05

- Calculate the probability of at least one generator working

We subtract the probability of all failing from 1: \( P(\text{at least one working}) = 1 - P(F) = 1 - 0.010648 = 0.989352 \).
06

- Interpret the result

The probability that at least one of the three backup generators works in case of a power failure is approximately \( 0.9894 \) or 98.94%. This appears to be a high reliability for the stadium's needs.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Backup Generators
Backup generators are essential for ensuring a stable power supply during emergencies. Large stadiums often use them to maintain operations if the main power source fails.

These generators kick in automatically when there is a power outage. Having multiple backup generators increases the chances that the stadium will still have power even if one or two generators don't work.

It's like having multiple lifelines; if one fails, the others can still save the day. But how reliable are these generators? Understanding their failure rates and how probabilities work is crucial for planning their usage effectively.
Failure Probability
In probability, failure rate is a key metric that tells us how often a system or component will fail. For backup generators, this rate is crucial to assess their reliability.

Let's consider the exercise: each generator has a 22% chance of failing when needed. This means that out of 100 times, a single generator will fail 22 times on average. So, the probability that one generator works is \(P(W) = 1 - 0.22 = 0.78\). This gives us a 78% chance that any given generator will operate successfully during a power failure.

Understanding this probability helps stadium managers evaluate how many generators they might actually need to ensure continuous power.
Multiplication Rule
The multiplication rule in probability helps us find the likelihood of multiple independent events all happening. Here, we use it to calculate the probability of all three generators failing at the same time.

Since each generator has a 22% chance of failing independently, the combined probability of all three failing is \((0.22)^3\), also written as \( P(F) = (0.22)^3\).

Applying the multiplication rule, this equals about 0.010648, or roughly 1.065%. This small probability indicates that it's quite unlikely for all three generators to fail simultaneously.
Redundancy in Systems
Redundancy means having extra components to improve system reliability. In the context of stadium backup generators, redundancy ensures power availability even if one or more generators fail.

If we have three generators, the probability that at least one works can be calculated by subtracting the probability of all failing from one: \(P(\text{at least one working}) = 1 - P(F)\).

With \(P(F)\) being 0.010648, we get \(P(\text{at least one working}) = 1 - 0.010648 = 0.989352\). This high probability (approximately 98.94%) shows how effective redundancy is in emergency systems, making it highly reliable for the stadium's needs.

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