Ch. 4 - Probability

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 4 - Probability

Problem 4.2.31a

Chapter 4, Problem 4.2.31a

Surge Protectors Refer to the accompanying figure showing surge protectors p and q used to protect an expensive television. If there is a surge in the voltage, the surge protector reduces it to a safe level. Assume that each surge protector has a 0.985 probability of working correctly when a voltage surge occurs.

a. If the two surge protectors are arranged in series, what is the probability that a voltage surge will not damage the television? (Do not round the answer.)

Verified step by step guidance

Step 1: Understand the arrangement of the surge protectors in series. When two components are arranged in series, both must work correctly for the television to be protected. This means the probability of protection is the product of the probabilities of each surge protector working correctly.

Step 2: Identify the probability of each surge protector working correctly. From the problem, each surge protector has a probability of 0.985 of working correctly during a voltage surge.

Step 3: Use the formula for the probability of both surge protectors working correctly in series: $ P_{series} = P_{p} \times P_{q} $, where $ P_{p} $ and $ P_{q} $ are the probabilities of surge protectors p and q working correctly.

Step 4: Substitute the given probabilities into the formula: $ P_{series} = 0.985 \times 0.985 $. This calculation will give the probability that a voltage surge will not damage the television.

Step 5: Interpret the result. The calculated probability represents the likelihood that both surge protectors will function correctly in series, ensuring the television is protected from a voltage surge.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability of Independent Events

In probability theory, independent events are those whose outcomes do not affect each other. For example, if two surge protectors operate independently, the probability of both functioning correctly during a voltage surge can be calculated by multiplying their individual probabilities. This concept is crucial for determining the overall effectiveness of multiple protective devices in series.

Series Configuration

In a series configuration, devices are connected in such a way that the failure of one device affects the entire system. For surge protectors, this means that if one fails to work, the voltage surge can damage the television. Understanding this configuration is essential for calculating the overall probability of protection when multiple devices are used.

Complementary Probability

Complementary probability refers to the likelihood of an event not occurring, which is calculated as 1 minus the probability of the event occurring. In the context of surge protectors, to find the probability that the television is not damaged, one must consider the probability that at least one surge protector fails and subtract it from 1. This concept is vital for accurately assessing the risk of damage.

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Complementary Events

Related Practice

Textbook Question

Corporate Officers and Committees The Self Driving Unicycle Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO), and chief human resources officer (CHR). It must also appoint a strategic planning committee with five different members. There are 15 qualified candidates, and officers can also serve on the committee.

a. How many different ways can the five officers be appointed?

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Textbook Question

Redundancy in Computer Hard Drives It is generally recognized that it is wise to backup computer data. Assume that the following refer to use of Western Digital model WD60EFRX hard drives, which have an annual failure rate of 3.66% (based on data from Backblaze, Inc.).

a. If you store all of your computer data on a single hard drive, what is the probability that the drive will fail during a year?

Textbook Question

In Exercises 9–20, use the data in the following table, which lists survey results from high school drivers at least 16 years of age (based on data from “Texting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,” by O’Malley, Shults, and Eaton, Pediatrics, Vol. 131, No. 6). Assume that subjects are randomly selected from those included in the table. Hint: Be very careful to read the question correctly.

Texting While Driving If two of the high school drivers are randomly selected, find the probability that they both texted while driving.

a. Assume that the selections are made with replacement. Are the events independent?

Textbook Question

Design Your Own Lottery You have been given the task of creating a new lottery. For each \$1 ticket, the player will select 6 different numbers from 1 to 25 (without replacement), and the only prize will be the jackpot won by players who select the six numbers (in any order) that are later drawn.

a. What is the probability of winning with one ticket?

Textbook Question

Mega Millions As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers from 1 to 70 and, in a separate drawing, you must also select the correct single number from 1 to 25.

a. Find the probability of winning the jackpot.

Textbook Question

In Exercises 21-28, find the probability and answer the questions.

X-Linked Genetic Disease Men have XY (or YX) chromosomes and women have XX chromosomes. X-linked recessive genetic diseases (such as juvenile retinoschisis) occur when there is a defective X chromosome that occurs without a paired X chromosome that is not defective. In the following, represent a defective X chromosome with lowercase x, so a child with the xY or Yx pair of chromosomes will have the disease and a child with XX or XY or YX or xX or Xx will not have the disease. Each parent contributes one of the chromosomes to the child.

a. If a father has the defective x chromosome and the mother has good XX chromosomes, what is the probability that a son will inherit the disease?