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

Question

The Centers for Disease Control (CDC) estimates that 0.6% of Americans have AIDS. The HIV-1 tests correctly diagnose the presence of HIV in 95% of the persons who have it and correctly diagnose its absence in 92% of the persons who don't.

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

Answer 1

Let's think of the CORRECT positive diagnoses P(+ | POS) and the CORRECT absence diagnoses as P(+ | NEG).  Also, we'll neglect 0.6% of sample size information and, instead, assume a 100% accurate determination of who actually has HIV, and who doesn't, as the basis in this testing scenario [P(POS) = 1 for an accurate base rate] and we'll let (+) denote a true, or accurate, test result.

We now consider Bayes' Theorem to determine the probability of a person listed as testing positive is indeed CORRECTLY testing positive:

P(person | +)  =  P(+ | POS) * P(POS) / [ P(+ | POS) * P(POS)  +  P(+ | NEG) * P(NEG) ]

P(person | +)  =  0.95  / [ (0.95 + ((1 - 0.92) * (1 - 0.006) ] 

=  0.95 / ( 0.95 + 0.07952 )   =    0.95 / 1.02952

=  0.92276  or   ~  92.3 % 

 

Therefore, the likelihood that a positive test result would be incorrect (i.e. should be negative) is  100 % - 92.3 %  =  roughly 7.7 %

https://en.wikipedia.org/wiki/Probability

* Also, consider the article on Bayes' Theorem in the 'Drug testing' section:

https://en.wikipedia.org/wiki/Bayes%27_theorem#Drug_testing