Bayes' Theorem tells you how to update a probability when new evidence arrives. It is the engine behind medical diagnostic interpretation, spam filters, machine learning classifiers, and rational belief revision. This guide covers the formula and what each piece (prior, likelihood, posterior) means, the intuition behind updating beliefs with evidence, a derivation from the definition of conditional probability, the classic rare-disease screening problem and why even doctors get it wrong, real applications to spam filters and court cases, common mistakes like the base rate fallacy and prosecutor's fallacy, and worked examples ending with a note on Bayesian vs frequentist thinking.