A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
Bartleby Related Questions Icon

Related questions

bartleby

Concept explainers

Topic Video
Question
### Probability Analysis of Disease Detection Using a Laboratory Blood Test

A laboratory blood test has a 95% effectiveness rate in detecting a certain disease when it is actually present. However, the test also yields a “false positive” result for 1% of healthy individuals tested. Given that 0.5% (or 0.005) of the population actually has the disease, we want to determine the probability that a person has the disease given that the test result is positive.

To solve this, we use Bayes' theorem, which incorporates conditional probabilities. Here is a breakdown of the relevant probabilities:

1. **Probability of a positive test given the disease is present (True Positive Rate):** 
   - \( P(\text{Positive} | \text{Disease}) = 0.95 \)

2. **Probability of a positive test given the disease is not present (False Positive Rate):**
   - \( P(\text{Positive} | \text{No Disease}) = 0.01 \)

3. **Probability of having the disease (Prevalence):**
   - \( P(\text{Disease}) = 0.005 \)

4. **Probability of not having the disease:**
   - \( P(\text{No Disease}) = 1 - P(\text{Disease}) = 1 - 0.005 = 0.995 \)

We are interested in finding \( P(\text{Disease} | \text{Positive}) \), the probability of having the disease given a positive test result.

According to Bayes' theorem:

\[ 
P(\text{Disease} | \text{Positive}) = \frac{P(\text{Positive} | \text{Disease}) \cdot P(\text{Disease})}{P(\text{Positive})} 
\]

Where \( P(\text{Positive}) \) is the total probability of a positive test result, which can be found using the law of total probability:

\[ 
P(\text{Positive}) = P(\text{Positive} | \text{Disease}) \cdot P(\text{Disease}) + P(\text{Positive} | \text{No Disease}) \cdot P(\text{No Disease}) 
\]

Plugging in the numbers:

\[ 
P(\text{Positive}) = (0.95 \cdot 0.005) + (0.01 \cdot 0.995)
expand button
Transcribed Image Text:### Probability Analysis of Disease Detection Using a Laboratory Blood Test A laboratory blood test has a 95% effectiveness rate in detecting a certain disease when it is actually present. However, the test also yields a “false positive” result for 1% of healthy individuals tested. Given that 0.5% (or 0.005) of the population actually has the disease, we want to determine the probability that a person has the disease given that the test result is positive. To solve this, we use Bayes' theorem, which incorporates conditional probabilities. Here is a breakdown of the relevant probabilities: 1. **Probability of a positive test given the disease is present (True Positive Rate):** - \( P(\text{Positive} | \text{Disease}) = 0.95 \) 2. **Probability of a positive test given the disease is not present (False Positive Rate):** - \( P(\text{Positive} | \text{No Disease}) = 0.01 \) 3. **Probability of having the disease (Prevalence):** - \( P(\text{Disease}) = 0.005 \) 4. **Probability of not having the disease:** - \( P(\text{No Disease}) = 1 - P(\text{Disease}) = 1 - 0.005 = 0.995 \) We are interested in finding \( P(\text{Disease} | \text{Positive}) \), the probability of having the disease given a positive test result. According to Bayes' theorem: \[ P(\text{Disease} | \text{Positive}) = \frac{P(\text{Positive} | \text{Disease}) \cdot P(\text{Disease})}{P(\text{Positive})} \] Where \( P(\text{Positive}) \) is the total probability of a positive test result, which can be found using the law of total probability: \[ P(\text{Positive}) = P(\text{Positive} | \text{Disease}) \cdot P(\text{Disease}) + P(\text{Positive} | \text{No Disease}) \cdot P(\text{No Disease}) \] Plugging in the numbers: \[ P(\text{Positive}) = (0.95 \cdot 0.005) + (0.01 \cdot 0.995)
Expert Solution
Check Mark
Knowledge Booster
Background pattern image
Probability
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
Text book image
A First Course in Probability (10th Edition)
Probability
ISBN:9780134753119
Author:Sheldon Ross
Publisher:PEARSON
Text book image
A First Course in Probability
Probability
ISBN:9780321794772
Author:Sheldon Ross
Publisher:PEARSON