Probability
Basics
Probability measures the likelihood of an event: \(P(A)=\frac{|A|}{|\Omega|}\) for equally likely outcomes.
Rules
| Property | Statement |
|---|---|
| Addition | \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) |
| Complement | \(P(A^c)=1-P(A)\) |
| Multiplication | \(P(A\cap B)=P(A)P(B|A)\) |
Conditional Probability
Bayes' theorem: \(P(A|B)=\frac{P(B|A)P(A)}{P(B)}\)
Examples
Example 1. A fair die is rolled. What is P(even | > 2)?
Solution. Even and >2: {{4,6}}, so P = 2/4 = 1/2.
Deep Dive: Probability
This section builds durable understanding of probability in statistics through definition-first reasoning, theorem mapping, and error-checking workflows.
Use a two-pass method: first derive the structure symbolically, then validate with a concrete numerical or geometric test case.
Visual Intuition
Convert algebra into a diagram, graph, or dependency map before solving. Visual-first analysis reduces sign errors and makes assumptions explicit.
Practice Set
Practice A. Re-derive one key formula on this page from first principles and annotate each transformation.
Target. Your final line should include assumptions, derivation path, and a quick verification.
Practice B. Build an application scenario using probability and solve it with both symbolic and numeric methods.
Target. Compare outputs and explain any approximation gap.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Editorial update: Reviewed on 2026-04-14 for notation consistency, conceptual clarity, and exercise quality.