Bayes' Theorem as Belief Updating

Start with prevalence

Bayes' theorem combines a prior probability with evidence. A prior is the probability before seeing new evidence. A likelihood describes how expected that evidence is under each possibility. The posterior is the updated probability after considering the evidence.

Suppose only 1 in 1000 jobs contains a serious defect. A detector catches 99% of defective jobs but falsely alerts on 1% of clean jobs. In 100,000 jobs, expect 100 defects: about 99 alert. Of 99,900 clean jobs, about 999 also alert. Among roughly 1,098 alerts, only 99 are real - about 9%, not 99%.

posterior defect among alerts
= true defect alerts / all alerts
= 99 / (99 + 999) ≈ 9%

Scenario: An operator hears the detector is 99% accurate and assumes every alert is almost certainly real. Low base prevalence means false alerts can still dominate. The prior matters.

Tip: Build a table for a concrete population of 1,000 or 100,000 before using the formula. Counts make base-rate effects much harder to overlook.