Module 2 · Probability Foundations · Topic 8
Intuition. Bayes’ theorem lets you reverse conditional probabilities. If you know P(B | A) and want P(A | B), Bayes’ theorem connects them using the prior probability P(A) and the total probability P(B).
P(A | B) = P(B | A) × P(A) / P(B)
P(A | B) = P(A and B) / P(B) — Probability of A after restricting attention to cases where B occurred.P(A | B) = P(B | A)P(A) / P(B) — Updates a prior probability after observing evidence.A disease affects 1% of people. A test is 95% sensitive (positive when sick) and 90% specific (negative when healthy). P(sick | positive) = (0.95 × 0.01) / ((0.95 × 0.01) + (0.10 × 0.99)) = 0.0095 / 0.1085 ≈ 8.8%.
The medical test example shows why rare diseases lead to many false positives. Always consider the base rate (prior probability) - ignoring it is one of the most common probability mistakes.
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