Margin of error for imbalanced discrete set

I’m evaluating the performance of a classifier regarding its false negatives. The classifier performed over 9090 samples, from which 9000 were labeled as negative. I randomly chose 800 samples (out of the 9000) and found that 4 were wrong. This means a 0.5% false negative rate. I also know that the precision (in all the 90 samples labeled positive) is 10%. I want to answer questions like:

• What’s the probability that the false negative rate in all the 9000 labeled negative samples is greater than 2%?
• What’s the margin of error for a 95% confidence?