/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Large stadiums rely on backup ge... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 22

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 that at least one generator works is approximately 98.94%. This appears to be adequate for the stadium's needs.

Step by step solution

01

- Understanding the Problem

The problem involves finding the probability that at least one out of three backup generators will work during a power failure, given that each generator fails with a probability of 22%.
02

- Calculate the Probability of Generator Failure

Since each generator fails 22% of the time, the probability of a single generator working is 1 - 0.22 = 0.78 or 78%.
03

- Calculate the Probability that All Generators Fail

We need to find the probability that all three generators fail. This is the product of the individual failure probabilities: \[ P(\text{All Fail}) = 0.22 \times 0.22 \times 0.22 = 0.22^3 = 0.010648 \]
04

- Calculate the Probability of At Least One Generator Working

The probability that at least one generator works is the complement of the probability that all three generators fail: \[ P(\text{At Least One Works}) = 1 - P(\text{All Fail}) = 1 - 0.010648 = 0.989352 \]
05

- Evaluate the Result

The probability that at least one generator works is 0.989352, or about 98.94%. This is a high probability, indicating that the backup generators are very reliable for the stadium's needs.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability Calculation
Calculating probability is fundamental in understanding real-world scenarios. In this exercise, we are dealing with the probability that backup generators will work during a power failure. A probability measures the chance of an event occurring and ranges from 0 (impossible) to 1 (certain). Here, we are interested in the probability that at least one of the three generators will work, given a known failure rate.
The main points are:
  • Each generator has a failure probability of 22%, or 0.22.
  • Therefore, the probability it works is the complement: 1 - 0.22 = 0.78.
Breaking down complex probability calculations into steps makes them less daunting and more manageable.
Complement Rule
In probability, the complement rule helps us find the probability of an event by subtracting the probability of its complement from 1. Here, the complement of the event 'at least one generator works' is 'all generators fail'.
Understanding the complement rule is key for this problem:
  • The probability of 鈥榓ll three generators failing鈥� is calculated by multiplying individual failure probabilities: \( P(\text{All Fail}) = 0.22 \times 0.22 \times 0.22 = 0.022^3 = 0.010648 \).
  • Using the complement rule, the probability that 鈥榓t least one generator works鈥� is: \( 1 - P(\text{All Fail}) \).
  • This step-by-step complements logical reasoning in calculating the desired probability.
Generator Failure Rate
The generator failure rate is an essential input for our probability calculations. In the given problem, each generator has a failure rate of 22%.
This percentage expresses how often the generator fails on average. Here鈥檚 how it affects our calculations:
  • Failure rate of 22%, or 0.22, means the generator works 78% of the time (\( 1 - 0.22 = 0.78 \)).
  • For three generators to fail concurrently, the individual failure probabilities multiply: \( P(\text{All Fail}) = 0.22 \times 0.22 \times 0.22 = 0.010648 \).
These failure rates are fundamental in assessing the reliability of the stadium's backup power system.
Stadium Power Backup
Backup generators are critical for large stadiums to ensure continuous operations during power failures. Ensuring reliability in such situations is paramount.
For the stadium to securely operate, the probability calculations help determine whether at least one generator will work when needed:
  • After calculations, the probability of having at least one generator work is approximately 0.989352 or 98.94%.
  • Such a high probability indicates strong reliability in the backup system, essential for large venues hosting significant events.
Providing a reliable power backup system ensures safety and uninterrupted service, highlighting the importance of these probability assessments.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

While this exercise was being created, Weather.com indicated that there was a \(60 \%\) chance of rain for the author's home region. Based on that report, which of the following is the most reasonable interpretation? a. \(60 \%\) of the author's region will get rain today. b. In the author's region, it will rain for \(60 \%\) of the day. c. There is a \(0.60\) probability that it will rain somewhere in the author's region at some point during the day.

Find the probability and answer the questions. In a Pew Research Center survey of Internet users, 3732 respondents say that they use social networking sites and 1380 respondents say that they do not use social networking sites. What is the probability that a randomly selected person does not use a social networking site? Does that result suggest that it is unlikely for someone to not use social networking sites?

A study addressed the issue of whether pregnant women can correctly predict the gender of their baby. Among 104 pregnant women, 57 correctly predicted the gender of their baby (based on data from "Are Women Carrying 'Basketballs'...," by Perry, DiPietro, Constigan, Birth, Vol. 26, No. 3). If pregnant women have no such ability, there is a \(0.327\) probability of getting such sample results by chance. What do you conclude?

Find the probabilities and indicate when the "5\% guideline for cumbersome calculations" is used. In a study of helicopter usage and patient survival, results were obtained from 47,637 patients transported by helicopter and 111,874 patients transported by ground (based on data from "Association Between Helicopter vs Ground Emergency Medical Services and Survival for Adults with Major Trauma" by Galvagno et al., Journal of the American Medical Association, Vol. 307, No. 15). a. If 1 of the 159,511 patients in the study is randomly selected, what is the probability that the subject was transported by helicopter? b. If 5 of the subjects in the study are randomly selected without replacement, what is the probability that all of them were transported by helicopter?

High Fives a. Five "mathletes" celebrate after solving a particularly challenging problem during competition. If each mathlete high fives each other mathlete exactly once, what is the total number of high fives? b. If \(n\) mathletes shake hands with each other exactly once, what is the total number of handshakes? c. How many different ways can five mathletes be seated at a round table? (Assume that if everyone moves to the right, the seating arrangement is the same.) d. How many different ways can \(n\) mathletes be seated at a round table?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.