### 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)

Name: ### Probability Analysis of Disease Detection Using a Laboratory Bloo
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: ### 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

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

Math

Probability

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...

See similar textbooks

Similar questions

Mathew is pleased to see a positive difference in the pilot sample, but Russell is cautious about committing a Type I error. What is the Type I error, and what is its probability of occurrence?
Say you’re a chemical engineer studying whether a certain trace impurity is present in aproduct. An experiment has a probability of 0.80 of detecting the impurity if it is present. Theprobability of not detecting the impurity if it is absent is 0.90. You have reason to believe that40% of products actually contain the impurity. You perform an experiment which detected animpurity – what is the probability that the impurity is not really there?
A manufacturer of aspirin claims that the proportion of headache sufferers who get relief with just two aspirins is 64%. What is the probability that in a random sample of 580 headache sufferers, less than 59.9% obtain relief? (Please Do not forget to check certain conditions.)
A diagnostic test for disease X correctly identifies the disease 95% of the time. False positives occur 7%. It is estimated that 5.13% of the population suffers from disease X. Suppose the test is applied to a random individual from the population. What is the percentage chance that, given a negative result, the person does not have disease X? What is the percentage chance that, the person will be misclassified?
A new medical test has been designed to detect the presence of a certain disease. Among those who have the disease, the probability that the disease will be detected by the new test is 0.74. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is 0.04. It is estimated that 14 % of the population who take this test have the disease.If the test administered to an individual is positive, what is the probability that the person actually has the disease?
An experiemental drug has been shown to be 75% effective in eliminating symptoms of allergies in animals studies. A small human study involve 6 participants is conducted. What is the probability that the drug is effective on half of the participants
A test to determine whether a certain antibody is present is 99.7% effective. This means that the test will accurately come back negative if the antibody is not present (in the test subject) 99.7% of the time. The probability of a test coming back positive when the antibody is not present (a false positive) is 0.003. Suppose the test is given to seven randomly selected people who do not have the antibody. What is the probability that the test comes back negative for all sevenpeople? and what is the probability that the test comes back positive for at least one of the seven people?
Apparently, depression significantly increases the risk of developing dementia later in lift (BBC News, July 6, 2010). In a recent study it was reported that 22% of those who had depression went on to develop dementia, compared to only 17% of those who did not have depression. Suppose 10% of all people suffer from depression. What is the probability of a person developing dementia? If a person has developed dementia, what is the probability that the person suffered from depression earlier in life?
Radio frequency identification (RFID) is an electronic scanning technology that can be used to identify items in a number of ways. One advantage of RFID is that it can eliminate the need to manually count inventory, which can help improve inventory management. The technology is not infallible, however, and sometimes errors occur when items are scanned. The probability that a scanning error occurs is 0.0069. 1. Find the probability that the number of items scanned incorrectly is between 16 and 20 , inclusive, from the next 5100 items scanned. *Kindly explain the procedure in Excel for this* A) The probability is (Round to four decimal places as needed.) 2. Find the expected number of items incorrectly from the next 5100 items B) The probability is (Round to four decimal places as needed.)
Assume that a student alarm clock has a 6.6 % daily failure rate. What is the probability that the students alarm clock ill not work on the day of an important final exam?