Question

Elisa Test

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 99.8% of

the time and will come back positive for the presence of

HIV antibodies 0.2% of the time (a false positive). If an in-

dividual has the HIV virus, the test will come back positive

99.8% of the time and will come back negative 0.2% of the

time (a false negative). Approximately 0.7% of the world

population has the HIV virus.

(a) What is the probability that a randomly selected individ-

ual has a test that comes back positive?

(b) What is the probability that a randomly selected individual

has the HIV virus if the test comes back positive?

Answer 1

We are given here that:
P( -ve | no HIV) = 0.998, P( + ve | no HIV) = 0.002

Also, we are given here that:
P( +ve | HIV) = 0.998 and therefore P(-ve | HIV) = 0.002

Also we are given that Approximately 0.7% of the world population has the HIV virus, therefore
P(HIV) = 0.007

a) Using law of total probability we get here:

P(+) = P(+ | HIV)P(HIV) + P(+ | no HIV)P(no HIV) = 0.998*0.007 + 0.002*(1 - 0.007) = 0.008972

Therefore 0.008972 is the required probability here.

b) Given that the test is positive, probability that the person has HIV is computed using Bayes theorem here as:

P(HIV | +ve) = P(+ | HIV)P(HIV) / P(+)

= 0.998*0.007 / 0.008972

= 0.7786

Therefore 0.7786 is the required probability here.