Probability forms the mathematical foundation of statistical inference, enabling us to quantify uncertainty and make rigorous predictions from data. It operates within a formal framework where events in a sample space are assigned numerical measures between 0 and 1, governed by Kolmogorov's axioms on a σ-algebra. Understanding probability is essential not only for modeling randomness but also for grasping how distributions, expectations, and limit theorems converge to make statistical methods work. A key insight: probability bridges the gap between theoretical models and real-world variability — and mastering its language from first principles unlocks all of modern statistics and machine learning.
What This Cheat Sheet Covers
This topic spans 17 focused tables and 152 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
Table 1: Foundational Concepts
| Concept | Example | Description |
|---|---|---|
(Ω, F, P) triple | The complete formal framework: triple (Ω, F, P) where Ω is the sample space, F is the σ-algebra of events, and P is the probability measure. | |
S = {H, T} for coin flip | • The set of all possible outcomes of a random experiment • denoted Ω or S. | |
F = {∅, {H}, {T}, {H,T}} | • Collection of subsets closed under complement and countable unions • defines which events can be assigned probabilities. | |
A = {2, 4, 6} for even die | • A subset of the sample space • can be simple (one outcome) or compound (multiple outcomes). | |
P(A) ≥ 0P(S) = 1P(A ∪ B) = P(A) + P(B) if disjoint | Kolmogorov's three axioms: non-negativity, normalization to 1, and countable additivity for mutually exclusive events. |