Q: Everyone at GSU takes a test for a rare disease (only present in 0.2% of the population). The test has a false positive rate of 1%. There is never a false negative. You get a positive result on the test. What is the probability you have the disease?
Q: Everyone at GSU takes a test for a rare disease (only present in 0.2% of the population). The test has a false positive rate of 1%. There is never a false negative. You get a positive result on the test. What is the probability you have the disease?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Question:** Everyone at GSU takes a test for a rare disease (only present in 0.2% of the population). The test has a false positive rate of 1%. There is never a false negative. You get a positive result on the test. What is the probability you have the disease?
**Explanation:** This scenario involves calculating the probability using Bayes' Theorem, which takes into account the prevalence of the disease, the accuracy of the test, and the occurrence of false positives.
1. **Prevalence of the Disease:** The disease is present in 0.2% of the population.
2. **False Positive Rate:** The probability that the test is positive when you do not have the disease is 1%.
3. **False Negative Rate:** There is no occurrence of false negatives in this test scenario.
The goal is to determine the probability you actually have the disease given that you tested positive. This is calculated as follows:
\[
P(\text{Disease | Positive}) = \frac{P(\text{Positive | Disease}) \times P(\text{Disease})}{P(\text{Positive})}
\]
Where:
- \( P(\text{Positive | Disease}) = 1 \) (since there are no false negatives)
- \( P(\text{Disease}) = 0.002 \)
- \( P(\text{Positive | No Disease}) = 0.01 \)
- \( P(\text{No Disease}) = 0.998 \)
To find \( P(\text{Positive}) \):
\[
P(\text{Positive}) = P(\text{Positive | Disease}) \times P(\text{Disease}) + P(\text{Positive | No Disease}) \times P(\text{No Disease})
\]
The detailed computation is left as an exercise for students to practice applying Bayes' Theorem in real-world scenarios.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b8f61c3-b57f-4943-8538-4e0823d8c6e6%2Fabdaca39-4e26-46a5-a31e-ea1daee486a6%2Fuhnil1_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:** Everyone at GSU takes a test for a rare disease (only present in 0.2% of the population). The test has a false positive rate of 1%. There is never a false negative. You get a positive result on the test. What is the probability you have the disease?
**Explanation:** This scenario involves calculating the probability using Bayes' Theorem, which takes into account the prevalence of the disease, the accuracy of the test, and the occurrence of false positives.
1. **Prevalence of the Disease:** The disease is present in 0.2% of the population.
2. **False Positive Rate:** The probability that the test is positive when you do not have the disease is 1%.
3. **False Negative Rate:** There is no occurrence of false negatives in this test scenario.
The goal is to determine the probability you actually have the disease given that you tested positive. This is calculated as follows:
\[
P(\text{Disease | Positive}) = \frac{P(\text{Positive | Disease}) \times P(\text{Disease})}{P(\text{Positive})}
\]
Where:
- \( P(\text{Positive | Disease}) = 1 \) (since there are no false negatives)
- \( P(\text{Disease}) = 0.002 \)
- \( P(\text{Positive | No Disease}) = 0.01 \)
- \( P(\text{No Disease}) = 0.998 \)
To find \( P(\text{Positive}) \):
\[
P(\text{Positive}) = P(\text{Positive | Disease}) \times P(\text{Disease}) + P(\text{Positive | No Disease}) \times P(\text{No Disease})
\]
The detailed computation is left as an exercise for students to practice applying Bayes' Theorem in real-world scenarios.
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