Correct :)

But one easy way to understand this problem is that, with only 95% precision, that hospital is extremely lousey :D ... Consider me, an idiot as a doctor, and I always make the following verdict:
no matter who is the patient, I always immediately claim he is HIV negative.
My clinical accuracy is 99.9% (only that 1/1000 true HIV positives will be mistaken.). Way much higher than that lousey hospital.

BTW, an easy way to improve the performance of hospital is to repeat the exams. But that means, you will probably have higher danger of getting infected by HIV positives, given the fact that you have to test your blood again and again at that lousey place :D