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Calculate conditional probability P(A|B) using Bayes' theorem. Common for medical testing, spam filtering, and decision making.
Scenario: A disease affects 1% of the population. A test has 90% sensitivity (correctly identifies sick people) and 5% false positive rate.
Question: If you test positive, what's the probability you actually have the disease?
With P(A)=0.01, P(B|A)=0.90, P(B|¬A)=0.05:
P(A|B) ≈ 15.4%
Even with a positive test, there's only ~15% chance of having the disease! This is why screening requires confirmatory tests.
When the prior probability P(A) is low (rare event), even accurate tests can have low predictive value. This is the "base rate fallacy" - we often ignore how rare the condition is.
Medical diagnosis, spam filters, search engines, machine learning, weather forecasting, legal reasoning, and any situation where you update beliefs based on new evidence.