The Centers for Disease Control...

Question

The Centers for Disease Control estimates that 0.3% of Americans have aids. The HIV-1 tests correctly diagnose the presence of HIV in 95.5% of the persons who have it and correctly diagnose its absence in 92.5% of the persons who don't.

 

Find the probability that a person whose result is negative actually is negative.

Answer 1

So we know that 0.3% of people have aids, which can also be interpreted as 99.7% of people who don't. 

 

So the number of people who will get a "negative" on their tests are
People who have HIV but the test fails and says they don't HIV
People who don't have HIV and the test succeeds and says they don't have HIV

 

If the tests correctly diagnose the presence of HIV in 95.5% of people who have it, this means that of that 0.3% of the population, this test will work 95.5% of the time. This means that 4.5% of those who have the disease will test negative, also known as a false negative. This is one part of our targeted population. 

 

If the tests corrently diagnose the absence of HIV in 92.5% of the people who don't have it, this means that for 99.7% of the population, this test will work 92.5% of the time, which is a true negative (they don't have it, and the test said they dont have it). 

 

Going back to those those two targeted populations, we can calculate the total number of people who got negatives (sum the two groups) and the total number of people who got true negatives (latter of the group).

 

In this situation, it would look like this

 

true negative / [false negative + true negative]

 

or

 

(0.925 * 0.997) / [(0.045 * 0.003) + (0.925 * 0.997)]