The Problem:
There is a new test that will test for the HIV virus, but we are not sure of whether this test's results are usually correct or not. We are given the following information:
$ P (+ | HIV) = 0.9, P (- | HIV) = 0.1\! $
$ P (+ | no HIV) = 0.1, P (- | no HIV) = 0.9\! $
We are also given that only $ 0.5%\! $ of the population has the HIV virus. The rest do not. Are the results usually correct, and what can you tell from the results?
The Solution:
First, we determine the probability that a random person within the population tests positive. Remember, this person will be selected at random, so we have no clue whether or not he/she actually has the virus or not.
$ P(+) = P (+ \cap HIV) + P(+ \cap no HIV)\! $
From previous notes in this class, we know that the following is true:
$ P (+ \cap HIV) = P (+ | HIV) P (HIV)\! $ and $ P(+ \cap no HIV) = P (+ | no HIV) P (not HIV)\! $
Thus, we can conclude that:
$ P(+) = P (+ | HIV) P (HIV) + P (+ | no HIV) P (not HIV)\! $
This can easily be solved with the given information.
$ P (+ | HIV) P (HIV) = 0.9 (0.005)\! $ and $ P (+ | no HIV) P (not HIV) = 0.1 (0.995)\! $
We can add both of those products together to obtain the probability that our random person tests positive, which turns out to be a $ 0.104\! $ probability. From this, we can already kind of tell that our test is not that great because $ 0.5%\! $ of the people have HIV, but a much larger number of people will test positive.
