if the test comes back negative, it’s wrong only about 3 times out of 10,000,”

So there-in lies the problem, which is that things like cancer, are already rare. Say there are 6 real instances of cancer in 10,000. Say the system says 20 are cancer, 3 actually are, and it missed 3. That means it was only positively correct at a rate of 50%, even though its overall accuracy is well over 99%