/*! 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 31 The Associated Press (San Luis O... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 31

The Associated Press (San Luis Obispo Telegram-Tribune, August 23,1995 ) reported the results of a study in which schoolchildren were screened for tuberculosis (TB). It was reported that for Santa Clara County, California, the proportion of all tested kindergartners who were found to have TB was 0.0006 . The corresponding proportion for recent immigrants (thought to be a high-risk group) was \(0.0075 .\) Suppose that a Santa Clara County kindergartner is to be selected at random. Are the events selected student is a recent immigrant and selected student has \(T B\) independent or dependent events? Justify your answer using the given information.

Short Answer

Expert verified
Due to the lack of specific information needed, it's inconclusive whether the events 'selected student is a recent immigrant' and 'selected student has TB' are independent or dependent events.

Step by step solution

01

Understand Probabilities

First, understand that a probability can be equated to the given proportions. Using the information provided, the probability that a randomly chosen kindergartner has TB, denoted as P(TB), is 0.0006. The probability that a randomly selected kindergartner is a recent immigrant and has TB, denoted as P(Immigrant and TB), is 0.0075.
02

Understand Independence

Events are said to be independent if the probability of both occurring is the product of the probabilities of each occurring. In other words, P(A and B) = P(A) * P(B) for independent events A and B. Here, we are asked to determine if the events 'selected student is a recent immigrant' and 'selected student has TB' are independent.
03

Apply Rule of Independence

If the events were independent, then the probability that a kindergarten student is a recent immigrant and has TB would be the product of the probabilities of each event. Based on our understanding of how the probabilities are set up in this situation, there is not enough information provided to calculate the probability that a kindergartner is a recent immigrant, required for the independence check. Therefore, we cannot say for certain whether these events are independent or dependent.
04

Conclude answer

With the provided information, we cannot conclusively determine whether the events 'selected student is a recent immigrant' and 'selected student has TB' are independent or dependent events. The specific probability of a kindergartner being a recent immigrant is needed to accomplish this.

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.

Independence of Events
When we talk about events in probability being independent, we mean that the occurrence of one event does not affect the occurrence of another.
For example, when flipping a fair coin, getting a "heads" on one flip does not affect whether the next flip will also be "heads."
This is because the flips are independent events.
  • To determine independence, we check if the probability of both events happening together is the same as the two events happening separately.
  • Mathematically, for events A and B to be independent, the equation must hold: \[ P(A \text{ and } B) = P(A) \times P(B) \]
In our exercise, we are seeking to understand if having TB and being a recent immigrant are independent by determining if their joint occurrence is simply the product of their individual probabilities.
This would mean that being a recent immigrant does not influence the chance of having TB, and vice versa.
Dependent vs Independent Events
In the realm of probability, dependent and independent events describe how two events interact with each other.
If events are dependent, the occurrence of one affects the other.
Conversely, for independent events, the occurrence of one does not impact the other.
  • Dependent Events: Imagine pulling two consecutive cards from a deck without replacement. If your first card is an ace, there are now fewer aces in the deck, and this affects the probability of drawing another ace.
  • Independent Events: Rolling a pair of dice is independent. The result of one die does not affect the result of the other.
In our exercise, we were asked to consider if a student being a recent immigrant and having TB are independent.
To do this, we would need more information, particularly the individual probability of a kindergartner being a recent immigrant in the area.
Without this data, we cannot make a definitive claim about their dependence or independence.
Calculating Probabilities
Calculating probabilities allows us to measure the chance of an event occurring.
To find the probability of a single event, we often use observed or theoretical frequencies.
  • A single event probability, like \( P(TB) = 0.0006 \), represents the chance that any kindergartner has TB.
  • A joint probability, such as \( P(\text{Immigrant and } TB) = 0.0075 \), shows the likelihood of two events happening at the same time.
For events to be independent, their joint probability should equal the product of their individual probabilities.
In our problem, while we know some probabilities, we lack the specific probability of a kindergartner being a recent immigrant from which we could confirm independence.
When calculating, always remember:
  • Add individual probabilities for mutually exclusive events.
  • Multiply probabilities for independent events.
  • Understand the context to ensure accurate calculations.

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

The article "Anxiety Increases for Airline Passengers After Plane Crash" (San Luis Obispo Tribune, November 13,2001 ) reported that air passengers have a 1 in 11 million chance of dying in an airplane crash. This probability was then interpreted as "You could fly every day for 26,000 years before your number was up." Comment on why this probability interpretation is misleading.

The following statement is from a letter to the editor that appeared in USA Today (September 3,2008 ): "Among Notre Dame's current undergraduates, our ethnic minority students \((21 \%)\) and international students \((3 \%)\) alone equal the percentage of students who are children of alumni (24\%). Add the \(43 \%\) of our students who receive need-based financial aid (one way to define working-class kids), and more than \(60 \%\) of our student body is composed of minorities and students from less affluent families." Do you think that the statement that more than \(60 \%\) of the student body is composed of minorities and students from less affluent families is likely to be correct? Explain.

The student council for a school of science and math has one representative from each of five academic departments: Biology (B), Chemistry (C), Mathematics (M), Physics (P), and Statistics (S). Two of these students are to be randomly selected for inclusion on a university-wide student committee. a. What are the 10 possible outcomes? b. From the description of the selection process, all outcomes are equally likely. What is the probability of each outcome? c. What is the probability that one of the committee members is the statistics department representative? d. What is the probability that both committee members come from laboratory science departments?

A large cable company reports that \(42 \%\) of its customers subscribe to its Internet service, \(32 \%\) subscribe to its phone service, and \(51 \%\) subscribe to its Internet service or its phone service (or both). a. Use the given probability information to set up a "hypothetical \(1000 "\) table. b. Use the table to find the following: i. the probability that a randomly selected customer subscribes to both the Internet service and the phone service. ii. the probability that a randomly selected customer subscribes to exactly one of the two services.

A student placement center has requests from five students for employment interviews. Three of these students are math majors, and the other two students are statistics majors. Unfortunately, the interviewer has time to talk to only two of the students. These two will be randomly selected from among the five. a. What is the sample space for the chance experiment of selecting two students at random? (Hint: You can think of the students as being labeled \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},\) and \(\mathrm{E}\). One possible selection of two students is \(\mathrm{A}\) and \(\mathrm{B}\). There are nine other possible selections to consider.) b. Are the outcomes in the sample space equally likely? c. What is the probability that both selected students are statistics majors? d. What is the probability that both students are math majors? e. What is the probability that at least one of the students selected is a statistics major? f. What is the probability that the selected students have different majors?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.