This may be a very simple question, but perhaps that's why I'm worried I'm not doing it correctly!
We're running a study that follows volunteers once per month (where a month is a four week period, or 28 days). These volunteers are at high risk of acquiring HIV infection, and are counseled and tested regularly. We are screening for acute HIV infection, which is a brief (estimated at 10 days) period during which the volunteer is negative by standard tests, but positive by testing for genetic material. We call this antibody negative, PCR positive.
If we assume that no one misses any study visits, is the chance of detecting a volunteer during this period of acute HIV infection essentially 10/28, or 36%?
You have to assume that the chance of getting an infection on each day is equal. This may not be true for example if you always test on a Monday (you said 28 days so you always test on the same day of the week) but they are more likely to get an infection over the weekend due to their weekend activities. But if it is true, then yes, $10/28$ is the answer.
Also, keep in mind that there is a lot of variability in the time until antibodies are detectable depending on the patient. But if it really is 10 days on average across all patients, and there is no bias in your sample when you say your "at risk" population, then your answer is still fine.