### Probability Analysis of Disease Detection Using a Laboratory Blood TestA 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)

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Bayes' Theorem: Let A1, A2, ... , An be n mutually exclusive and exhaustive events with P(Ai) > 0…

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A laboratory blood test is 95 % effective in detecting a certain disease when it is actually present. However, the test also yields a “false positive" result for 1 % of the healthy persons tested. If .5 % of the population actually has the disease, what is the probability a person has the disease given the test result is positive.

A laboratory blood test is 95 % effective in detecting a certain disease when it is actually present. However, the test also yields a “false positive" result for 1 % of the healthy persons tested. If .5 % of the population actually has the disease, what is the probability a person has the disease given the test result is positive.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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