don't forgot the condition

Don't forgot the condition when you apply the theorem. You assumed that test results is independent of whether the person has HIV or not, which is not always true! AND I don't believe that a good hospital should use the overall accuracy as a criterion, that's why it's a lousy hospital, or that's the trick the hospital used to cheat people, one should not care much about the overall accuracy here, instead one should care other two independ values. If you rate a hospital using the overall accuracy rate, then you are a lousy statistician too! Unfortunately, I guess many companies are doing this to cheat regular people who don't understand statistics well.