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Coscott comments on Open thread, July 21-27, 2014 - Less Wrong
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Are we assuming the two tests are independent?
If so, the original cancer rate was 5:95. Multiply that by 80:9 for the likelihood ratio of getting a positive to get 400:855, which is ~30% as you said. Then, you multiply by the likelihood ratio of getting the second negative 20:91, to get 8000:77805, which as a probability is 8000/(8000+77805)~9.3%.