They were doing great until they got up to the re-test, claiming that the second positive gives you 99% certainty you have cancer. That only works if the second test is completely independent from the first. If you repeat the first test a second time, only for those who get a positive result on the first, the same condition that caused a false positive on the first can cause a false positive on the second.
In reality, a cheap blood or urine test is likely to be followed by a more expensive test on a second portion of the same sample, then by an even more expensive tissue biopsy. Redoing an identical test only reduces random errors. It does not address the diagnostic bias of the test itself.
For instance, a pregnancy test detects hCG, from the placenta of a developing fetus. A man with various types of cancer or liver disease may also produce hCG, and can therefore produce a false positive for every test of that type, no matter how many repetitions. This does not give him greater confidence that he is pregnant!
Understanding Bayes also requires an understanding of event independence! For truly independent events, P(x|y) = P(x) and P(y|x) = P(y).
In reality, a cheap blood or urine test is likely to be followed by a more expensive test on a second portion of the same sample, then by an even more expensive tissue biopsy. Redoing an identical test only reduces random errors. It does not address the diagnostic bias of the test itself.
For instance, a pregnancy test detects hCG, from the placenta of a developing fetus. A man with various types of cancer or liver disease may also produce hCG, and can therefore produce a false positive for every test of that type, no matter how many repetitions. This does not give him greater confidence that he is pregnant!
Understanding Bayes also requires an understanding of event independence! For truly independent events, P(x|y) = P(x) and P(y|x) = P(y).