Probability Theory

Probability Theory is the mathematical framework for quantifying uncertainty. A probability measure P assigns to each Event in a Sample Space S a number in [0, 1] satisfying Kolmogorov's three axioms: non-negativity, normalization, and countable additivity.

P(E) \ge 0

P(S) = 1

P\!\left(\bigcup_{i=1}^{\infty} E_i\right) = \sum_{i=1}^{\infty} P(E_i)

• The domain \mathcal{F} is the event algebra — the collection of subsets of S to which probabilities are assigned.
• Countable additivity applies only to mutually exclusive events E_i.
• Every other rule — the complement rule, addition rule, and Conditional Probability — is derived from these axioms.
• It provides the foundation for statistics, actuarial science, and stochastic modeling.

Verifying Probability Axioms for a Die Roll A fair six-sided die is rolled, so S = \{1, 2, 3, 4, 5, 6\} with each outcome having probability \tfrac{1}{6}. What is the probability of rolling a number greater than 4?

Answer The event E = \{5, 6\} consists of two mutually exclusive outcomes, so by countable additivity: P(E) = P(\{5\}) + P(\{6\}) = \frac{1}{6} + \frac{1}{6} = \frac{1}{3} As a check, P(E) + P(E^c) = \frac{1}{3} + \frac{2}{3} = 1 = P(S), consistent with the axioms.