SOLUTION: The standard test for the HIV virus is the ELISA test, which tests for the presence of HIV antibodies. If an individual does not have the HIV virus, the test will come back negativ

Question 1183274: The standard test for the HIV virus is the ELISA test, which tests for the presence of HIV antibodies. If an individual does not have the HIV virus, the test will come back negative for the presence of HIV antibodies 99.7% of the time and will come back positive for the presence of HIV antibodies 0.3% of the time (a false positive). If an individual has the HIV virus, the test will come back positive 99.7% of the time and will come back negative 0.3% of the time (a false negative). Approximately 0.67% of the world population has the HIV virus. What is the probability that a randomly selected individual has the HIV virus if the test comes back positive?

Answer by ikleyn(52777) (Show Source): You can put this solution on YOUR website!
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The standard test for the HIV virus is the ELISA test, which tests for the presence of HIV antibodies.
If an individual does not have the HIV virus, the test will come back negative for the presence
of HIV antibodies 99.7% of the time and will come back positive for the presence of HIV antibodies 0.3% of the time (a false positive).
If an individual has the HIV virus, the test will come back positive 99.7% of the time and will come back negative 0.3% of the time
(a false negative). Approximately 0.67% of the world population has the HIV virus.
What is the probability that a randomly selected individual has the HIV virus if the test comes back positive?
~~~~~~~~~~~

Let X be the entire population.


Then the number of whose who has the HIV virus is 0.067*X persons.


The number of those who has the HIV virus test positive is  


        N = (1 - 0.067)*0.003*X + 0.067*X*0.997


The first addend in the right side comes from those who has no HIV virus, and the second addend comes from those who has HIV virus.


The problem asks about the conditional probability

    
    P =  = 

      = cancel X in the numerator and denominator and continue = 

      =  = 0.9627 = 96.27%   (rounded).   ANSWER

Solved.

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