/*! 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 89 Bright lights? A string of Chris... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 89

Bright lights? A string of Christmas lights contains 20 lights. The lights are wired in series, so that if any light fails, the whole string will go dark. Each light has probability 0.02 of failing during a 3 -year period. The lights fail independently of each other. Find the probability that the string of lights will remain bright for 3 years.

Short Answer

Expert verified
The probability is approximately 0.6676.

Step by step solution

01

Identify the Probability of Success for One Light

Each light has a 0.02 probability of failing. Therefore, the probability of a light not failing (or remaining bright) for 3 years is the complement: \( P( ext{not failing}) = 1 - 0.02 = 0.98 \).
02

Determine Total Probability of Success for All Lights

Since the lights are wired in series, all 20 lights need to remain bright for the string to stay lit. The probability that all lights will remain bright is the product of the individual probabilities (assuming independence): \( (0.98)^{20} \).
03

Calculate the Probability

Compute the probability using the formula: \( (0.98)^{20} \), which results in approximately 0.6676. This is the probability that the string of Christmas lights will remain bright for 3 years.

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.

Understanding Independent Events
In probability theory, independent events are events where the outcome of one event does not influence or change the outcome of another event. This is crucial when calculating the probability of multiple events happening sequentially. For instance, if you roll a die two times, the result of the first roll doesn't affect the second roll. They are independent events.
For the Christmas lights problem, each light bulb is considered an independent event. That means whether one light fails or not doesn't alter the probability of another light failing. This independence allows us to calculate the probability of all the lights remaining functional by simply multiplying the probability of each individual light staying bright over the specified duration.
Concept of Series Circuit in Bulbs
A series circuit connects multiple electrical components along a single path. In this setup, if there's a break at any point in the circuit (like a burned-out bulb), the entire circuit stops working, which is different from a parallel circuit where other paths might continue working.
In the given exercise, all 20 Christmas lights are connected in a series circuit. If any one of these bulbs were to fail, it would cause the entire string to go out. That's why it's important to ensure each bulb remains lit for the whole string to stay bright. The design exemplifies a series circuit dependency, where the failure of just one component affects the entire system.
Applying the Complement Rule
The complement rule in probability helps to find the probability of an event not occurring by subtracting the probability of the event occurring from 1. So, if the probability of a bulb failing is 0.02, the probability of the bulb not failing (or remaining operational) is the complement, calculated as:
  • \( P(\text{not failing}) = 1 - P(\text{failing}) = 1 - 0.02 = 0.98 \)
This formula is key in determining the probability of success for independent events. Once we know the chance of one light bulb operating well, we can then calculate the overall probability of all 20 bulbs operating, which in this case is
  • \((0.98)^{20}\)
By understanding the complement rule, you ascertain the safety margin of each component and ensure the entire system has a proactive support calculation for remaining functional over the given timeframe.

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

In the United Kingdom's Lotto game, a player picks six numbers from 1 to 49 for each ticket. Rosemary bought one ticket for herself. She had the lottery computer randomly select the six numbers. When the six winning numbers were drawn, Rosemary was surprised to find that none of these numbers appeared on the Lotto ticket she had bought. Should she be? Design and carry out a simulation to answer this question. Follow the four-step process.

Crawl before you walk ( 3.2 ) At what age do babies learn to crawl? Does it take longer to learn in the winter, when babies are often bundled in clothes that restrict their movement? Perhaps there might even be an association between babies' crawling age and the average temperature during the month they first try to crawl (around six months after birth). Data were collected from parents who brought their babies to the University of Denver Infant Study Center to participate in one of a number of studies. Parents reported the birth month and the age at which their child was first able to creep or crawl a distance of 4 feet within one minute. Information was obtained on 414 infants \((208\) boys and 206 girls). Crawling age is given in weeks, and average temperature (in \({ }^{\circ} \mathrm{F}\) ) is given for the month that is six months after the birth month. $$\begin{array}{lcc}\hline & \text { Average } & \text { Average } \\\\\text { Birth month } & \text { crawling age } & \text { temperature } \\\\\text { January } & 29.84 & 66 \\\\\text { February } & 30.52 & 73 \\\\\text { March } & 29.70 & 72 \\\\\text { April } & 31.84 & 63 \\\\\text { May } & 28.58 & 52 \\\\\text { June } & 31.44 & 39 \\\\\text { July } & 33.64 & 33 \\\\\text { August } & 32.82 & 30 \\\\\text { September } & 33.83 & 33 \\\\\text { 0ctober } & 33.35 & 37 \\\\\text { November } & 33.38 & 48 \\\\\text { December } & 32.32 & 57 \\\\\hline\end{array}$$ Analyze the relationship between average crawling age and average temperature. What do you conclude about when babies learn to crawl?

Rolling dice Suppose you roll two fair, six-sided dice-one red and one green. Are the events "sum is \(7 "\) and "green die shows a 4 " independent? Justify your answer.

Taking the train According to New Jersey Transit, the 8: 00 A.M. weekday train from Princeton to New York City has a \(90 \%\) chance of arriving on time. To test this claim, an auditor chooses 6 weekdays at random during a month to ride this train. The train arrives late on 2 of those days. Does the auditor have convincing evidence that the company's claim isn't true? Design and carry out a simulation to estimate the probability that a train with a \(90 \%\) chance of arriving on time each day would be late on 2 or more of 6 days. Follow the four-step process.

Brushing teeth, wasting water? A recent study reported that fewer than half of young adults turn off the water while brushing their teeth. Is the same true for teenagers? To find out, a group of statistics students asked an SRS of 60 students at their school if they usually brush with the water off. In the sample, 27 students said "No." The Fathom dotplot below shows the results of taking 200 SRSs of 60 students from a population in which the true proportion who brush with the water off is 0.50 . (a) Explain why the sample result does not give convincing evidence that fewer than half of the school's students brush their teeth with the water off. (b) Suppose instead that 18 students in the class's sample had said "No." Explain why this result would give strong evidence that fewer than \(50 \%\) of the school's students brush their teeth with the water off.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.