Bayes Worked Example
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Bayes reasoning updates a starting rate with new evidence, and the clearest way to run it is to take a concrete population and count the people in each group.
A condition affects one percent of people. A test catches ninety-nine percent of those who have it and wrongly flags five percent of those who do not. Someone tests positive. Take ten thousand people. One percent have it, so one hundred do and nine thousand nine hundred do not. Of the hundred who have it, the test catches ninety-nine. Of the nine thousand nine hundred who do not, five percent are wrongly flagged, which is four hundred and ninety-five people. Now count everyone holding a positive result: ninety-nine plus four hundred and ninety-five, which is five hundred and ninety-four. Of those, ninety-nine actually have the condition. Ninety-nine divided by five hundred and ninety-four is about seventeen percent. A positive result on a test this accurate, for a condition this rare, is more likely to be a false alarm than a finding.
The base rate does the work here, and it is the piece people discard first. Reasoning from the test's accuracy alone yields something near ninety-nine percent, which is wrong by a factor of five — because a small error rate applied to a very large healthy group still outnumbers a large success rate applied to a tiny sick one.
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