/*! 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 57 A study considered risk factors ... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

A study considered risk factors for HIV infection among intravenous drug users [11] . It found that \(40 \%\) of users who had \(\leq 100\) injections per month (light users) and \(55 \%\) of users who had \(>100\) injections per month (heavy users) were HIV positive. What is the probability that at least 4 of the 20 users are HIV positive?

Short Answer

Expert verified
Assuming the worst case, the probability that at least 4 out of 20 users are HIV positive is approximately 0.9752.

Step by step solution

01

Define the Problem

We need to calculate the probability that at least 4 out of 20 users are HIV positive. We have different probabilities for light and heavy users, but we aren't provided with how many are of each type in our sample of 20 users.
02

Assumption Consideration

Let's assume the worst-case scenario for the highest probability: all 20 users are heavy users, where the probability of being HIV positive is 0.55. This simplifies our calculation since we can use the binomial probability formula.
03

Use the Binomial Distribution

The binomial distribution formula is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n = 20 \), \( p = 0.55 \), and \( k \) is the number of users who are HIV positive. We need the cumulative probability for \( k \geq 4 \).
04

Calculate Probabilities for Fewer than 4 Positive

We need to calculate: \( P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \), hence \( P(X = k) = \binom{20}{k} (0.55)^k (0.45)^{20-k} \) for \( k = 0, 1, 2, 3 \).
05

Calculate Probabilities for Each \(k\)

- \( P(X = 0) = \binom{20}{0} (0.55)^0 (0.45)^{20} \)- \( P(X = 1) = \binom{20}{1} (0.55)^1 (0.45)^{19} \)- \( P(X = 2) = \binom{20}{2} (0.55)^2 (0.45)^{18} \)- \( P(X = 3) = \binom{20}{3} (0.55)^3 (0.45)^{17} \).
06

Calculate the Cumulative Probability

After calculating the probabilities in Step 5, sum up the values obtained to get the cumulative probability for \( P(X < 4) \).
07

Find the Complement Probability

The probability that at least 4 users are HIV positive is 1 minus the cumulative probability for fewer than 4 users: \( P(X \geq 4) = 1 - P(X < 4) \).
08

Compute the Final Answer

Using the calculated values in the previous steps, you find \( P(X \geq 4) \). After computing each term, sum them and subtract from 1 to find the final probability.

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.

HIV Infection Study
Understanding the risk factors for HIV infection among different groups is critical in public health. This involves studying various communities to determine which behaviors or characteristics increase the likelihood of contracting the virus. Among intravenous drug users, sharing needles is a known high-risk activity that can lead to a higher prevalence of HIV. This study, focusing on injection frequency, highlights a significant correlation between the number of injections and the risk of HIV infection.
  • Light users, defined as individuals with fewer than or equal to 100 injections per month, have a 40% rate of HIV positivity.
  • Heavy users, with more than 100 injections per month, show a 55% rate of HIV positivity.
These statistics underscore the importance of tailoring prevention efforts to high-risk groups by promoting safer practices and regular testing.
Probability Calculation
In probability studies, particularly when dealing with health-related events, it is essential to accurately calculate the likelihood of certain outcomes. For this HIV infection study among drug users, the aim is to determine the probability of at least 4 out of 20 users being HIV positive.
Probability calculations like these often utilize the binomial distribution, a statistical method that describes the probability of having a certain number of successes in a fixed number of trials. Here, a 'success' would be a user testing positive for HIV.
By assuming the highest infection probability (0.55), and employing the binomial formula, we calculate probabilities for different scenarios (0 to 3 users testing positive). The goal is to find the complement, which in this context means calculating 1 minus the cumulative probability of getting fewer than four positives.
Risk Factors Analysis
Analyzing risk factors involves identifying and understanding which behaviors contribute the most to the spread of disease such as HIV. For intravenous drug users, the frequency of injections is a key risk factor.
This study illustrates that as the number of monthly injections increases, so does the probability of being HIV positive. The analysis provides a clear connection between the level of drug use and the risk of infection, facilitating targeted interventions.
Such interventions could include:
  • Educational programs on safe needle practices.
  • Providing clean needles to reduce sharing.
  • Offering regular health check-ups and HIV testing.
Understanding these risk factors enables more effective public health strategies and could potentially lower infection rates.
Intravenous Drug Users Study
Researching specific populations, like intravenous drug users, sheds light on the dynamics of health risks within these groups. Such studies are vital for forming efficient, evidence-based public health policies.
This particular study divides the drug users into different categories based on their injection frequency, facilitating a clearer analysis of how behavior impacts health risks. Traditionally, these users are more challenging to study due to stigma and hidden behaviors, but their participation is essential.
The findings highlight the importance of a nuanced approach to public health, one that considers varying levels of risk within a community. Tailoring interventions more precisely to these groups' realities ensures better resource allocation and, ultimately, more successful health outcomes. By conducting studies like these, public health officials can better understand and mitigate high-risk behaviors.

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

One hypothesis is that gonorrhea tends to cluster in central cities. Suppose 10 gonorrhea cases are reported over a 3-month period among 10,000 people living in an urban county. The statewide incidence of gonorrhea is 50 per 100,000 over a 3 -month period. Is the number of gonorrhea cases in this county unusual for this time period?

Studies have been undertaken to assess the relationship between abortion and the development of breast cancer. In one study among nurses (the Nurses' Health Study II), there were 16,359 abortions among 2,169,321 person-years of follow-up for women of reproductive age. (Note: 1 personyear \(=1\) woman followed for 1 year.) What is the expected number of abortions among nurses over this time period if the incidence of abortion is 25 per 1000 women per year and no woman has more than 1 abortion?

A clinical trial was conducted among 178 patients with advanced melanoma (a type of skin cancer) (Schwartzentruber, et al. [18]). There were two treatment groups. Group A received Interleukin-2. Group B received Interleukin-2 plus a vaccine. Six percent of Group A patients and \(16 \%\) of group \(\mathrm{B}\) patients had a complete or partial response to treatment. Suppose we seek to extrapolate the results of the study to a larger group of melanoma patients. If 20 melanoma patients are given Interleukin-2 plus a vaccine, what is the probability that at least 3 of them will have a positive response to treatment? One issue is that some patients experience side effects and have to discontinue treatment. It was estimated that \(19 \%\) of patients receiving Interleukin-2 plus vaccine developed an arrhythmia (irregular heartbeat) and had to discontinue treatment. since the side effect was attributable to the vaccine, assume that the patients would continue to take Interleukin-2 if arrhythmia developed but not the vaccine and that the probability of a positive response to treatment would be the same as group \(A\) (i.e., \(6 \%\) ).

Evaluate \(9 !\)

An experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a \(10-\mathrm{mg}\) dose of the drug is lethal \(5 \%\) of the time within the first 4 hours; of the animals alive at 4 hours, \(10 \%\) will die in the next 4 hours. What is the probability that 0 rats will die in the 8-hour period?

See all solutions

Recommended explanations on Math Textbooks

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.