Subset of training set produces good results while full training set produces poor results
For the 200 positive samples, are there many different "types"? In other words, will future positive samples look nothing like the current positive samples? If this is the case, you may try anomaly detection in machine learning. 

From a lecture by Andrew Ng in Coursera, if you have small number of positive samples (e.g., 20), large number of negative samples (e.g., 10000), and it's hard for any algorithm to learn from the positive samples, you should use anomaly detection, otherwise, you can use supervised learning.