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Expected value is the one number that tells you whether a bet, investment, or decision is worth making in the long run. Most people have never calculated it. Here's how.

Before you can compute expected values, variances, or probability distributions, you need a way to assign numbers to uncertain outcomes. Random variables are that bridge — an apparently simple concept that makes all of statistical mathematics possible.

Most people update their beliefs wrong. Bayes' theorem gives the correct formula — and the gap between human intuition and Bayesian logic explains most bad reasoning about risk and evidence.

Every time you learn something new, the probability of everything else shifts. Conditional probability is the math that formalizes this — and it's behind Bayesian reasoning, medical testing, and every poker hand.

All of probability theory builds from three operations: AND (both events), OR (at least one), and NOT (neither). These rules have precise mathematical forms, and getting them right — especially the OR rule's overlap correction — is where beginners most often go wrong.

Pascal and Fermat invented probability to settle a gambling dispute. Three and a half centuries later, it underpins quantum mechanics, Bayesian reasoning, machine learning, and the entire statistical enterprise. What probability *is*, philosophically, remains surprisingly contested.

Probability starts with simple rules about how likelihoods combine — but those rules generate everything from the central limit theorem to machine learning. This is a grounded tour of the core framework: sample spaces, conditional probability, independence, and why Bayes' theorem keeps appearing.