I have this problem, and I have solved it but I'm confused about some points.
HIV infected approximately 1,500 people in country X last year1. Mark heard about HIV, so he made a test for the HIV. The test correctly identifies the presence of HIV in 80% of cases and only gives false positives in 1/10,0000 cases. In Mark's case, the test was positive for HIV and Mark is very concerned. Assume that in the population of X country there are 20 million people susceptible to HIV.
- What is the probability that Mark has HIV?
- The HIV virus is fatal in 4% of the cases. What is the probability that Mark will die this year? Assume a fatality rate of any cause (car accident, etc.) of 0.15% that is independent of whether or not Mark has HIV.
I believe that:
probability that Mark has HIV, P(HIV) = 1,500/20,000,000 = 0.000075
Hence, P(~HIV) = 1- 0.000075 = 0.999925
let Pos is test is positive
P(Pos|HIV) = 0.80
P(Pos|~HIV)= 1/10,0000 = 0.00001
For question 1, I believe it wants me to find P(Mark has HIV|test is positive)?
P(Mark has HIV|test is positive) = P(HIV|Pos)
P(HIV|Pos) = P(Pos|HIV).P(PHIV) / P(Pos)
= P(Pos|HIV).P(HIV) / P(Pos,HIV) + P(Pos,~HIV)
= P(Pos|HIV).P(HIV) / P(Pos|HIV).P(HIV) + P(Pos|~HIV).P(~HIV)
= 0.80 * 0.000075 / (0.80 * 0.000075 + 0.00001 * 0.999925)
For question 2:
the probability that Mark will die this year should be as follows: P(Mark has HIV|test is positive) * [P(Mark dies|Mark has HIV) + P(Mark dies|Accident)] + (1 - P(Mark has HIV|test is positive)) * P(Mark dies|Accident)
= 0.857 * (0.4 + 0.0015) + (1- 0.857) * 0.0015
For Question 1, I feel that I miss something. Am I?
For Question 2, my question is that do I have to calculate the probability of both cases if Mark has HIV and If he Doesn't have HIV.