/*! 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 1 Suppose that \(25 \%\) of people... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 1

Suppose that \(25 \%\) of people have a dog, \(29 \%\) of people have a cat, and \(12 \%\) of people own both. What is the probability that someone owns a dog or a cat?

Short Answer

Expert verified
The probability that someone owns a dog or a cat is 0.42.

Step by step solution

01

Convert Percentages to Probabilities

First, we need to convert the given percentages into decimal probabilities. Since 25% of people have a dog, the probability of owning a dog, \( P(D) \), is 0.25. Similarly, the probability of owning a cat, \( P(C) \), is 0.29, and the probability of owning both a dog and a cat, \( P(D \cap C) \), is 0.12.
02

Apply the Inclusion-Exclusion Principle

The probability that someone owns a dog or a cat can be calculated using the Inclusion-Exclusion Principle: \( P(D \cup C) = P(D) + P(C) - P(D \cap C) \). This principle prevents double-counting those who have both a dog and a cat.
03

Substitute Probabilities Into the Formula

Substitute the known probabilities into the formula: \( P(D \cup C) = 0.25 + 0.29 - 0.12 \). This step uses the numbers we derived from the percentages.
04

Perform the Calculation

Calculate \( P(D \cup C) \) by performing the arithmetic: \( 0.25 + 0.29 - 0.12 = 0.42 \). This result represents the probability of someone owning either a dog or a cat.

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.

Inclusion-Exclusion Principle
The inclusion-exclusion principle is a fundamental concept in probability theory that helps in calculating the probability of the union of multiple events. When determining the likelihood of two or more events occurring, this principle ensures that we do not double-count the scenarios where the events overlap.
For example, when calculating the probability of someone owning a dog or a cat, we have probabilities for individual events and the overlap. Here's how it works:
  • The probability of event A, such as owning a dog (P(D)), is considered.
  • The probability of event B, such as owning a cat (P(C)), is also considered.
  • The overlap, where both events occur, owning both a dog and a cat (P(D \cap C)), is subtracted.
Thus, the final formula becomes \( P(D \cup C) = P(D) + P(C) - P(D \cap C) \). This equation helps in accounting for the overlap, ensuring accuracy in probability calculations.
Percentages to Probabilities
Understanding how to convert percentages into probabilities is crucial because probability calculations require numbers between 0 and 1.
Percentages are often dealt with in everyday life, but probabilities quantify them in statistical terms.
  • To convert a percentage to a probability, simply divide the percentage by 100.
  • For example, if 25% of people own a dog, the probability \( P(D) \) is computed as \( \frac{25}{100} = 0.25 \).
  • Similarly, 29% translates to a probability \( P(C) \) of \(0.29\), and 12% to \( P(D \cap C) \) of \(0.12\).
Being comfortable with this conversion process ensures accuracy in any further steps of probability calculation. It's an essential precursor to applying the inclusion-exclusion principle correctly.
Probability Calculation
Probability calculation is the process of determining the likelihood of an event or a set of events. Once you have converted percentages to probabilities and understand the formula, you can calculate these probabilities accurately.
  • First, use any necessary predefined formulas, like the inclusion-exclusion principle, to relate multiple events.
  • Substitute the probabilities into the formula to set up your calculation. In our case, using the formula: \( P(D \cup C) = 0.25 + 0.29 - 0.12 \).
  • Perform the arithmetic simply, as shown: \( 0.25 + 0.29 - 0.12 = 0.42 \).
This result, \(0.42\), clearly shows the probability of someone owning a dog or a cat. Probability calculations, once steps are sequenced methodically, become straightforward. Mastery over each step ensures reliability in achieving correct results in statistical problems.

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

Recent research suggests that \(73 \%\) of Americans have a home phone, \(83 \%\) have a cell phone, and \(58 \%\) of people have both. What is the probability that an American has a) a home or cell phone? b) neither a home phone nor a cell phone? c) a cell phone but no home phone?

Dan's Diner employs three dishwashers. Al washes \(40 \%\) of the dishes and breaks only \(1 \%\) of those he handles. Betty and Chuck each wash \(30 \%\) of the dishes, and Betty breaks only \(1 \%\) of hers, but Chuck breaks \(3 \%\) of the dishes he washes. (He, of course, will need a new job soon. ...) You go to Dan's for supper one night and hear a dish break at the sink. What's the probability that Chuck is on the job?

A university requires its biology majors to take a course called BioResearch. The prerequisite for this course is that students must have taken either a Statistics course or a computer course. By the time they are juniors, \(52 \%\) of the Biology majors have taken Statistics, \(23 \%\) have had a computer course, and \(7 \%\) have done both. a) What percent of the junior Biology majors are ineligible for BioResearch? b) What's the probability that a junior Biology major who has taken Statistics has also taken a computer course? c) Are taking these two courses disjoint events? Explain. d) Are taking these two courses independent events? Explain.

A junk box in your room contains a dozen old batteries, five of which are totally dead. You start picking batteries one at a time and testing them. Find the probability of each outcome. a) The first two you choose are both good. b) At least one of the first three works. c) The first four you pick all work. d) You have to pick 5 batteries to find one that works.

After surveying 995 adults, \(81.5 \%\) of whom were over \(30,\) the National Sleep Foundation reported that \(36.8 \%\) of all the adults snored. \(32 \%\) of the respondents were snorers over the age of \(30 .\) a) What percent of the respondents were under 30 and did not snore? b) Is snoring independent of age? Explain.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.