What is Probability?
Probability is the mathematical language of uncertainty. It turns vague intuitions about chance into precise, consistent numbers.
Experiments & Outcomes
A random experiment is any process whose result cannot be predicted with certainty. The sample space Ω lists every possible outcome.
Events
An event is any subset of the sample space — a collection of outcomes we want to assign a probability to.
Union, Intersection, Complement
- A ∪ B — A or B (or both) occur
- A ∩ B — A and B both occur
- Ac — A does not occur
- Disjoint — A ∩ B = ∅, cannot overlap
What Does Probability Mean?
- Classical — ratio of favorable to equally-likely outcomes
- Frequentist — long-run relative frequency
- Bayesian — degree of belief, updated by evidence
Classical Probability
When every outcome is equally likely, probability is simply the ratio of favorable outcomes to total outcomes.
Frequentist View
Flip a coin a million times. The fraction of heads converges to 0.5. Probability is a property of a repeatable process, not a single trial.
Bayesian View
Probability measures degree of belief. It can be assigned to any uncertain proposition — even one-time events — and updated as evidence arrives.
Kolmogorov’s Axioms
- Axiom 1 — P(A) ≥ 0 for any event A
- Axiom 2 — P(Ω) = 1 (something must happen)
- Axiom 3 — P(A ∪ B) = P(A) + P(B) if A ∩ B = ∅
Complement & Addition
Two rules you will use in almost every probability calculation:
Permutations & Combinations
Classical probability requires counting outcomes. Order matters → permutations. Order irrelevant → combinations.
What you learned
Sample spaces, events, three interpretations, Kolmogorov’s axioms, the complement and addition rules, and counting methods — the complete foundation of probability.