/*! 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 7 Consider a family with a mother,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 7

Consider a family with a mother, father, and two children. Let \(A_{1}=\left\\{\text { mother has influenzal, } A_{2}=\\{ \text { father has influenzal, }\right.\) \(A_{3}=\left\\{\text { first child has influenzal, } A_{4}=\\{ \text { second child has influ- }\right.\) enzal, \(B=\) lat least one child has influenzal, \(C=\) lat least one parent has influenzal, and \(D=\\{\) at least one person in the family has influenzal.Express \(D\) in terms of \(B\) and \(C .\)

Short Answer

Expert verified
\(D = B \cup C\)

Step by step solution

01

Understand the problem and express D

The problem involves expressing the event "at least one person in the family has influenza" using the events "at least one child has influenza" and "at least one parent has influenza." This requires understanding that if either at least one child or at least one parent has influenza, then someone in the family has influenza.
02

Use set operations for expression

To express the event \( D \) in terms of \( B \) and \( C \), we use the union of these two events. In set theory, when we say at least one of several events occurs, we are referring to the union of these events. Therefore, \( D = B \cup C \).
03

Interpret the solution

The union \( B \cup C \) represents the occurrence that at least one event between children having influenza and parents having influenza is true, thereby covering all cases where anyone in the family has influenza. This satisfies the condition for \( D \).

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.

Probability Events
Probability events are fundamental to understanding biostatistics and many other fields that rely on statistical analysis. In this family-related problem, we're looking at specific events defined by who in the family might have influenza.
Each event is a statement about a particular subset of the family, such as "the mother has influenza" or "at least one child has influenza." These events have probabilities associated with how likely they are to occur.
  • Single Event: For example, the event that the mother has influenza, denoted by \(A_1\), is an individual probability event.
  • Compound Event: When considering "at least one child has influenza" (B) or "at least one parent has influenza" (C), we are dealing with compound probability events. These involve multiple scenarios adding together to fulfill the condition.
Understanding how these events relate helps us utilize probability theory to compute the likelihood of various health outcomes within the family.
Union of Sets
The union of sets in set theory helps describe situations where multiple events can occur. In our exercise, we must express the scenario "at least one person in the family has influenza" in terms of other defined events.
This involves the union of the set representing children with influenza (B) and parents with influenza (C). The union, represented by \(B \cup C\), means that if either event B or C happens, it's enough to meet the condition of our set D.
  • Union Operation: The operation is critical here. Set union incorporates all elements from both sets, providing a thorough understanding of all possible outcomes.
  • Set Representation: When using the union, imagine overlapping circles in a Venn diagram. The union includes every area that any circle covers.
This makes the set union a powerful way to simplify complex probability events like the ones dealing with family health statistics.
Family Health Statistics
Family health statistics often require us to consider various aspects of probability and set theory. By examining a family's health based on whether members have influenza, we apply these statistical methods to understand the bigger picture.
This particular exercise reflects real-life scenarios where multiple factors interact, showing how comprehensive health data is analyzed.
  • Analyzing Trends: When collecting family health statistics, it's important to determine how individual health events (like having the flu) impact overall family health.
  • Interpreting Data: Understanding relationships between events, like our events B and C, helps us spot trends and correlations that might suggest larger health patterns.
  • Preventive Measures: By analyzing these statistics, healthcare services can tailor preventive measures, optimize resource allocation, and make better health predictions for families.
Understanding family health statistics through probability and the union of sets equips us with tools to analyze complex medical data effectively, leading to improved healthcare decisions.

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

Consider a family with a mother, father, and two children. Let \(A_{1}=\left\\{\text { mother has influenzal, } A_{2}=\\{ \text { father has influenzal, }\right.\) \(A_{3}=\left\\{\text { first child has influenzal, } A_{4}=\\{ \text { second child has influ- }\right.\) enzal, \(B=\) lat least one child has influenzal, \(C=\) lat least one parent has influenzal, and \(D=\\{\) at least one person in the family has influenzal.Express \(C\) in terms of \(A_{1}, A_{2}, A_{3},\) and \(A_{4}\).

The ankle-arm blood-pressure index (AAl) is defined as the ratio of ankle systolic blood pressure/arm systolic blood pressure and is used for the diagnosis of lower extremity arterial disease. A study was conducted to investigate whether the AAl can be used as a screening test for atherosclerotic diseases in general [20]. The subjects were 446 male workers in a copper smelter in Japan. Each subject had an AAl determination as well as an electrocardiogram (ECG). From the ECG, an S-T segment depression was defined as an S-T segment \(\geq 0.1 \mathrm{mV}\) below the baseline in at least 1 of 12 leads in a resting ECG. S-T segment depression is often used as one characterization of an abnormal ECG. The data in Table 3.22 were presented relating AAl to S-T segment depression. Suppose the reproducibility of the AAl test were improved using better technology. Would the sensitivity of the test increase, decrease, or remain the same? why?

Suppose that a disease is inherited via an autosomal recessive mode of inheritance. The implications of this mode of inheritance are that the children in a family each have a probability of 1 in 4 of inheriting the disease.What is the probability that in a family with two children, both siblings are affected?

Suppose that a disease is inherited via a dominant mode of inheritance and that only one of the two parents is affected with the disease. The implications of this mode of inheritance are that the probability is 1 in 2 that any particular offspring will get the disease.Suppose the older child is affected. What is the probability that the younger child is affected?

The familial aggregation of respiratory disease is a wellestablished clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is \(.15 ;(2)\) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is \(.13 ;(3)\) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is \(.05 ;\) and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04.Suppose the smoking habits of the parents are independent and the probability that the mother is a current smoker is \(.4,\) whereas the probability that the father is a current smoker is .5. What is the probability that both the father and mother are current smokers?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

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.