91Ó°ÊÓ

Aerosolized vaccine for measles. An aerosolized vaccine for measles was developed in Mexico and has been used on more than 4 million childiren since 1980. Aerosolized vaccines have the advantages of being able to be administered by people without clinical training and do not cause injectionassociated infections. The percentage of children developing an immune response to measles after receiving the subcutaneous injection of the vaccine is \(95 \%\), and for those receiving the aerosolized vaccine, it is \(85 \%\). 9 There are 20 children to be vaccinated for measles using the aerosolized vaccine in a small rural village in India. We are going to count the number who developed an immune response to measles after vaccination. (a) Explain why this is a binomial setting. (b) What is the probability that at least one child does not develop an immune response to measles after receiving the aerosolized vaccine? What would be the probability that at least one child does not develop an immune response to measles if all children were vaccinated using the subcutaneous injection of the vaccine?

Step by step solution

01

Identifying Binomial Setting

In probability theory, a binomial setting is defined when there are more than one independent trials, each trial has only two possible outcomes (success or failure), and the probability of success is the same for each trial. Here, we are vaccinating 20 children with the aerosolized vaccine, each trial (child) has two outcomes (immune response/no immune response), and each child has an 85% chance of developing an immune response. Thus, this is a binomial setting.

02

Calculating Probability for Aerosolized Vaccine

First, we need to determine the probability that at least one child does not develop an immune response. This can be calculated by first finding the probability that all children develop an immune response, and then subtracting this from 1. The probability of all 20 children developing an immune response is given by \((0.85)^{20}\). Then subtract from 1 to find the probability that at least one child does not develop an immune response: \(1 - (0.85)^{20}\).

03

Calculating Probability for Subcutaneous Vaccine

For the subcutaneous vaccine, apply the same approach. The probability of developing an immune response for all 20 children is \((0.95)^{20}\). So, the probability that at least one child does not develop an immune response is \(1 - (0.95)^{20}\).

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 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory

Probability theory is a mathematical framework used to predict the likelihood of events. It is essential for statistical analysis and everyday decision-making. At its core, it involves understanding and calculating chances, which are expressed as probabilities.
In this exercise, probability theory is used to analyze the outcomes of a vaccination study. Each child's response to the vaccine represents a separate probability trial. The concept we are dealing with is a **binomial distribution**. This means that the situation involves several trials (vaccinations), each with two possible outcomes, like success (develop immune response) or failure (no immune response). Here, success is defined by the child developing an immune response.

• Independent Trials: Each child's immune response is independent of the others.
• Two Possible Outcomes: Success (immune response) or failure (no immune response).
• Constant Probability: The probability of success is constant for each trial (85% for aerosolized, 95% for subcutaneous).

This binomial setting allows us to calculate the likelihood of different outcomes using the binomial probability formula. This is crucial in understanding how many children will develop immunity.

Vaccination Study

In this scenario, the vaccination study focuses on the use of an aerosolized versus a subcutaneous vaccination for measles. These vaccines aim to trigger an immune response in children, protecting them from the disease.
Aerosolized vaccines offer several benefits:

• Easy Administration: They can be given by individuals without clinical training.
• Safer Delivery: They avoid risks like needle-related infections.

However, the study highlights that only 85% of children develop an immune response compared to the 95% success rate of the subcutaneous vaccine.
This difference poses significant implications for public health strategies, especially in remote areas. By analyzing the vaccination study through probability, we can better understand the trade-offs between ease of administration and effectiveness, allowing for data-driven decisions regarding vaccination programs.

Statistical Setting

The statistical setting of the problem is essential to understanding its analysis. Here, we use statistical methods to examine the vaccination's impact on the children's immune response.
The study employs a binomial distribution, which is a powerful statistical tool for analyzing experiments where outcomes fall into two categories. With 20 trials (children), each representing a chance of success (immune response) or failure (no immune response), we calculate probabilities for two settings:

• Aerosolized Vaccine: Each child has an 85% chance of showing an immune response.
• Subcutaneous Vaccine: Each child has a 95% chance of showing an immune response.

In both cases, we aim to find the likelihood that at least one child does not develop an immune response. This is essential in evaluating the vaccine's effectiveness and reliability.
The statistical analysis helps compare the performance of the two vaccination methods, guiding decision-makers on choosing the best approach based on their situation.

Immune Response Analysis

Analyzing the immune response is crucial in evaluating the vaccine's effectiveness. In this study, we focus on how many children develop immunity after vaccination.
Calculating the probability that at least one child does not develop an immune response involves subtracting the probability that all children develop the response from 1. Using the binomial probability formula for each method:

• Aerosolized Vaccine: The probability that all 20 children develop a response is \((0.85)^{20}\), therefore, the probability that at least one does not is \1 - (0.85)^{20}\.
• Subcutaneous Vaccine: The probability that all 20 children develop a response is \((0.95)^{20}\), thus, the probability that at least one does not is \1 - (0.95)^{20}\.

These calculations help illuminate how likely it is for the vaccines to fail in producing immunity. For public health, knowing this likelihood ensures better planning and management of vaccination programs to maximize immunity and minimize measles outbreaks.