Question in Description

Question

The Centers for Disease Control (CDC) estimates that 0.8% of Americans have AIDS. The HIV-1 tests correctly diagnose the presence of HIV in 99% of the persons who have it and correctly diagnose its absence in 93% of the persons who don't.

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

Answer 1

Let x be the population of the United States.

 

Then x/125 of the population has AIDS and 124x/125 of the population does not have AIDS.

 

If a person has AIDS, then then the HIV-1 test indicates that the person does not have AIDS at 1%, for the

whole population that has AIDS this represents (1/100)*(x/125) = x/12500

 

If a person does not have AIDS, then the HIV-1 test indicates that the person does not have AIDS at 93%, for the

whole population that does not have AIDS this represents (93/100)*(124x/125) = 11532x/12500

 

The test indicates that a person does not have AIDS by the sum of these number, 11533x/12500

 

Then the probability that a person whose result is negative actually is negative is

(11532x/12500)/(11533x/12500) = 11532/11533