II · Probability & randomness
Bayes theorem
What it is
P(A∣B) = P(B∣A) · P(A) / P(B). The rule for updating beliefs given evidence.
Where it lives
Spam filters, medical tests, A/B test priors, every Bayesian inference engine.
The key insight
Counterintuitive when P(A) is small: a 99%-accurate test on a 1-in-1000 disease still gives ~9% true-positive rate.