Interesting Probability

Tests used to diagnose medical condition are often imperfect, and give false positive or false negative results. A fairly cheap blood test for the Human Immunodeficiency Virus (HIV) that causes AIDS (Acquired Immune Deficiency Syndrome) has the following characteristics: the false negative rate is 2% and the false positive rate is 0.5%. It is assumed that around .04% of Canadian males are infected with HIV. Find the probability that if a male tests positive for HIV, he actually has HIV.

Solution: Suppose a male is randomly selected from the population, and define the events
A= {selected male has HIV}
B= {blood test is positive}
We are asked to find P(A/B). From the information given we know that
P(A)=.0004
p(!A)=.9996
P(B/A)=.98
P(B/!A)=.005
Therefore we can find
P(AB)=P(A)P(B/A)=.000392
P((!A)B)=P(!A)P(B/!A) =.004998
Therefore P(B)= P(AB)+P((!A)B)=.00539
and P(A/B)=P(AB)/P(B)=.0727
Thus, if a randomly selected male tests positive, there is still only a small probability (.0727) that they actually have HIV.

Find this on my stat textbook, I never thought of HIV or any other tests this way.