This article totally messed up their interpretation of the statistics. From some anonymous source they learned that the false positive rate of the test was 1/1000... so far, so good. But they then say that the test is "wrong about a positive result only about once in a thousand times". This is not equivalent to the previous statement. A false positive rate of 1/1000 means that about 1 in every 1000 cows tested will show up positive whether they have the disease or not. But, if a cow does show up as positive, the chances the test is wrong could be almost anything (9/10, 99/100, whatever), depending on the prevalence of the disease. If, as the article states, the rate of disease is much less than the rate of false positives, then most of the positives will be false, and the probability that a positive is false will be much greater than 1/2, and not the "1/1000" the article claims.
Just had to get that off my chest