Probability Theory

Probability is the mathematical language of uncertainty. It provides the axiomatic foundation on which all of statistical inference rests: without probability, there is no hypothesis testing, no confidence intervals, no Bayesian updating, no regression. This skill covers the core machinery from sample spaces through the central limit theorem.

Agent affinity: bayes (conditional probability, Bayes' theorem), pearson (distributional theory), efron (computational probability)

Concept IDs: stat-probability-foundations, stat-experimental-theoretical, stat-expected-value, stat-conditional-probability

Axioms and Sample Spaces

Kolmogorov's axioms

A probability function P on a sample space S satisfies:

1. Non-negativity: P(A) >= 0 for every event A.
2. Normalization: P(S) = 1.
3. Countable additivity: For mutually exclusive events A_1, A_2, ..., P(A_1 union A_2 union ...) = P(A_1) + P(A_2) + ...

Everything in probability follows from these three axioms plus set theory.