I am having much difficulty with this problem.
EIA tests are used to screen blood specimens for the presence of antibodies to HIV, the virus that causes AIDS. Antibodies indicate the presence of the virus. The test is quite accurate but is not always correct. Here are approx. probabilities of positive and negative EIA outcomes when the blood tested does and does not actually contain antibodies to HIV.
positive negative
Antibodies present .9985 .0015
Antibodies absent .006 .994
Suppose that 1% of a large population carries antibodies to HIV in their blood.
1) what is the probability that the EIA is positive for a randomly chosen person from this population?
2) what is the probability that a person has an antibody, given that the EIA test is positive?
Best Answer
Let $H$ be the event of a person having HIV and $P$ be a positive test. We know that $$ P(P|H)=0.9985\quad P(\bar P|H)=0.0015\\ P(\bar P|\bar H)=0.994\quad P(P|\bar H)=0.006\\ P(H)=0.01\quad P(\bar H)=0.99 $$ In question $1$, we are asked to calculate $P(P)$: \begin{align} P(P)&=P(P|H)P(H)+P(P|\bar H)P(\bar H)\\ &=0.9985\cdot 0.01+0.006\cdot 0.99\\ &=0.015385 \end{align} For the second question, we need $P(H|P)$. Using Bayes' rule: $$ P(H|P)=\frac{P(H)}{P(P)}P(P|H)=\frac{0.01}{0.015385}\cdot 0.9985=0.649009 $$ As you can see, the probability that you have HIV when tested positive is (only) $65%$.