/*! 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 113 According to a survey of 2000 ho... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 113

According to a survey of 2000 home owners, 800 of them own homes with three bedrooms, and 600 of them own homes with four bedrooms. If one home owner is selected at random from these 2000 home owners, find the probability that this home owner owns a house that has three or four bedrooms. Explain why this probability is not equal to \(1.0\)

Short Answer

Expert verified
The probability that a randomly selected homeowner from the survey owns a house with either three or four bedrooms is 0.7 or 70%.

Step by step solution

01

Identify the Total Outcomes

The total number of outcomes in an event is the total number of homeowners surveyed, which is 2000.
02

Identify Favorable Outcomes for Each Event

The favorable outcomes for corresponding events are: homeowners owning three-bedroom houses which is 800 and homeowners owning four-bedroom houses which is 600.
03

Calculate Individual Probabilities

Probability of a homeowner owning a three bedroom house is the ratio of homeowners with three-bedroom houses to total homeowners. Which is \(\frac{800}{2000}=0.4\). Similarly, probability of a homeowner owning a four bedroom house is \(\frac{600}{2000}=0.3\)
04

Add the Probabilities

To find the probability that a homeowner owns a house with either three or four bedrooms, we add the individual probabilities. So, Probability (three or four bedrooms) = Probability (three bedrooms) + Probability (four bedrooms) = 0.4 + 0.3 = 0.7.
05

Explain why the Total Probability is not 1

The total probability is not 1 (or 100%) because not all homeowners have either a three or four bedroom house. There are other homeowners who have either less than three or more than four bedrooms. Hence the combined probability is less than 1.

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.

Survey analysis
Understanding survey analysis helps us to make predictions based on collected data. In our exercise, a survey of 2000 homeowners gives us insights into housing preferences and distribution.

A survey collects responses to form a sample that reflects the larger population. In this case, we surveyed 2000 people to learn about their home sizes.
  • This survey aims to understand how many homes have three and four bedrooms.
  • The responses provide information on 800 homes with three bedrooms and 600 homes with four bedrooms.
This data is the foundation for calculating probabilities about housing features in the community.
Probability calculation
Probability tells us how likely an event is to happen. Calculating probability involves comparing favorable outcomes to the entire sample space.

In our problem, we need to find the likelihood that a randomly selected homeowner owns a house with either three or four bedrooms. This involves a few steps:
  • The total number of outcomes, or sample space, is the total number of people, which is 2000.
  • The favorable outcomes for owning a three-bedroom house is 800, and for owning a four-bedroom house is 600.
  • Individual probabilities are calculated as fractions: Probability for three bedrooms = \(\frac{800}{2000} = 0.4\) Probability for four bedrooms = \(\frac{600}{2000} = 0.3\)
These fractions represent the likelihood of picking a homeowner with a specific bedroom count from the survey.
Event outcomes
Events in probability refer to possible results from an experiment or survey. Here, the events are owning a three-bedroom home or a four-bedroom home.

When dealing with such events, it is important to focus on favorable outcomes:
  • Owning a three-bedroom home involves 800 favorable outcomes, which signifies 800 people fit this criteria.
  • Owning a four-bedroom home has 600 favorable outcomes, relating to 600 homeowners with such homes.
To find the probability of either event happening (a three or four-bedroom home), the probabilities are simply added together: \[0.4 + 0.3 = 0.7\] Thus, there is a 70% chance that a selected homeowner from the survey has a house with either three or four bedrooms.
Three and four bedrooms probability
Understanding why the total probability is not 1 is crucial in probability studies. When the probability is calculated, it accounts only for surveyed outcomes.

In our result, we found a probability of 0.7 for three or four bedrooms combined. However, this does not account for all possible outcomes (all possible house sizes) outside these measurements.
  • Some homeowners may have houses with less than three bedrooms.
  • Others may possess homes with more than four bedrooms.
This is why the probability is less than 1; it doesn鈥檛 cover all possibilities. The probability value of 0.7 implies that 70% of cases in these categories combined, while 30% remains in other categories not highlighted in the exercise.

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

A trimotor plane has three engines-a central engine and an engine on each wing. The plane will crash only if the central engine fails and at least one of the two wing engines fails. The probability of failure during any given flight is \(.005\) for the central engine and \(.008\) for each of the wing engines. Assuming that the three engines operate independently, what is the probability that the plane will crash during a flight?

The probability that a randomly selected college student attended at least one major league baseball game last year is .12. What is the complementary event? What is the probability of this complementary event?

Two thousand randomly selected adults were asked if they think they are financially better off than their parents. The following table gives the two- way classification of the responses based on the education levels of the persons included in the survey and whether they are financially better off, the same as. or worse off than their parents. $$ \begin{array}{lccc} \hline & \begin{array}{c} \text { Less Than } \\ \text { High School } \end{array} & \begin{array}{c} \text { High } \\ \text { School } \end{array} & \begin{array}{c} \text { More Than } \\ \text { High School } \end{array} \\ \hline \text { Better off } & 140 & 450 & 420 \\ \text { Same as } & 60 & 250 & 110 \\ \text { Worse off } & 200 & 300 & 70 \\ \hline \end{array} $$ Suppose one adult is selected at random from these 2000 adults. Find the following probabilities. a. \(P\) (better off or high school) b. \(P\) (more than high school or worse off) c. \(P(\) better off or worse off \()\)

A gambler has four cards - two diamonds and two clubs. The gambler proposes the following game to you: You will leave the room and the gambler will put the cards face down on a table. When you return to the room, you will pick two cards at random. You will win $$\$ 10$$ if both cards are diamonds, you will win $$\$ 10$$ if both are clubs, and for any other outcome you will lose $$\$ 10$$. Assuming that there is no cheating, should you accept this proposition? Support your answer by calculating your probability of winning $$\$ 10$$.

A random sample of 250 juniors majoring in psychology or communication at a large university is selected. These students are asked whether or not they are happy with their majors. The following table gives the results of the survey. Assume that none of these 250 students is majoring in both areas. $$ \begin{array}{lcc} \hline & \text { Happy } & \text { Unhappy } \\ \hline \text { Psychology } & 80 & 20 \\ \text { Communication } & 115 & 35 \\ \hline \end{array} $$ a. If one student is selected at random from this group, find the probability that this student is i. happy with the choice of major ii. a psychology major iii. a communication major given that the student is happy with the choice of major iv. unhappy with the choice of major given that the student is a psychology major v. a psychology major and is happy with that major vi. a communication major \(o r\) is unhappy with his or her major b. Are the events "psychology major" and "happy with major" independent? Are they mutually exclusive? Explain why or why not.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

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.