SOLUTION: In a region 4% of the population is thought to have a certain disease. A standard diagnostic test will correctly identify 92% of the people who have the disease. However, the tes

Algebra.Com
Question 189051: In a region 4% of the population is thought to have a certain disease. A standard diagnostic test will correctly identify 92% of the people who have the disease. However, the test also incorrectly diagnoses 10% of those who do not have the disease as having the disease. A randomly selected person in the region is tested for the disease.
What is the probability the test comes back positive?
What is the probability the test comes back positive and the person actually has the disease?
If the test comes back positive, what then is the conditional probability that he actually does have the disease?

Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
In a region 4% of the population is thought to have a certain disease. A standard diagnostic test will correctly identify 92% of the people who have the disease. However, the test also incorrectly diagnoses 10% of those who do not have the disease as having the disease. A randomly selected person in the region is tested for the disease.
------------
Use a tree diagram with d,d' for diseased and not diseased
Use p,n for diagnosed positive and diagnosed negative
---------------------------------------------------------------
What is the probability the test comes back positive?
P(p) = P(p and d) + P(p and d') = P(p}d)P(d) + P(p]d')P(d')
= 0.92*0.04 + 0.1*0.96 = 0.1328
==================================================================
What is the probability the test comes back positive and the person actually has the disease?
P(p and d) = P(p|d)*P(d) = 0.92*0.04 = 0.0368
==================================================================
If the test comes back positive, what then is the conditional probability that he actually does have the disease?
P(d}p) = P(d and p)/P(p) = 0.0368/[P(p and d) + P(p and d')
= 0.0368/0.1328 = 0.2771
===================================================================
Cheers,
Stan H.

RELATED QUESTIONS

A diagnostic test for a certain disease is said to be 90% accurate in that, if a person... (answered by Boreal)
A diagnostic test for a certain disease is said to be 90% accurate in that, if a person... (answered by VFBundy)
Consider a medical test for a certain disease. The probability of a person having the... (answered by Shin123)
If a test is carried out and it identifies a disease in 95% of people who have it. It... (answered by jim_thompson5910)
The proportion of people in a given community who have a certain disease is 0.004. A test (answered by robertb)
A medical test has been designed to detect the presence of a certain disease. Among those (answered by CPhill)
A certain medical test is known to detect 75% of the people who are afflicted with the... (answered by Boreal)
Suppose a test for a disease has a sensitivity of 95% and a specificity of 60%. Further... (answered by Boreal)
A new medical test has been designed to detect the presence of the mysterious... (answered by Boreal)