Axiomatic Probability and Addition Theorem

Lesson 19 of STA301 explores the modern mathematical formalization of probability developed by Andrey Kolmogorov in 1933, known as the Axiomatic Definition of Probability. Rather than relying on physical symmetry or hypothetical infinite trials, Kolmogorov defined probability as a real-valued set function P assigning a numerical measure to events in a sample space S according to three fundamental axioms: Axiom 1 (Non-negativity): For any event A, P(A) >= 0; Axiom 2 (Certainty): The probability of the sample space S equals unity, P(S) = 1; and Axiom 3 (Additivity): If A1, A2, ... are mutually exclusive events, then the probability of their union equals the sum of their individual probabilities: P(A1 U A2 U ...) = sum P(A_i). From these foundational axioms, several essential probability theorems are derived. First, the probability of the impossible event is zero: P(phi) = 0. Second, the Rule of Complementation establishes that the probability of the complementary event A' (not A) is P(A') = 1 - P(A), which simplifies computations when evaluating 'at least one' event occurrences. Third, if event A is a subset of event B (A subset B), then P(A) <= P(B). Fourth, the probability of any event is strictly bounded between zero and one: 0 <= P(A) <= 1. The lecture derives the Addition Theorem of Probability. For two mutually exclusive events A and B (where A cap B = phi), the theorem states: P(A U B) = P(A) + P(B). When events are not mutually exclusive (overlapping events with non-empty intersection A cap B != phi), the General Addition Theorem applies: P(A U B) = P(A) + P(B) - P(A cap B). Subtracting the joint intersection corrects for double-counting elements. The theorem extends to three events: P(A U B U C) = P(A) + P(B) + P(C) - P(A cap B) - P(A cap C) - P(B cap C) + P(A cap B cap C). In Virtual University midterm examinations, addition theorem derivations, complement calculations, and word problems featuring playing cards or dice are frequently tested. Reviewing sta301 handouts and practicing solved MCQs by Moaaz reinforces these principles.