Sounds right to me. Ok - so let's try to put some actual numbers in here for you. Lets pretend that 100000 people took this test. Now we know that 10% are actually HIV positive.

So that means that 10000 people are HIV positive. We also know the sensitivity is .999 which is the proportion of those 10000 people that are going to get a positive test back. So 10000*.999 = 9990 people have a positive test AND are HIV positive. Which means that 10 people have a negative test AND are HIV positive. Do you see why that last sentence is true?

We can do something similar for those people that are HIV negative. There are 90000 HIV negative people out of the 100000 (do you see why this is?) and we have a specificity of .9999. Specificity is the proportion of those that are actually HIV negative that get a negative test. So .9999*90000 = 89991 people are HIV negative AND have a negative test. Can you work out how to get how many people are HIV negative and have a positive test?

So from this we should be able to make a table

[Dason's table here]

Can you see how to answer the original question using this table?

Just so you know I'm going to head to bed so I won't be answer any more questions until morning. Good luck!