Classical and Relative Frequency Probability

Lesson 18 of STA301 investigates the philosophical and mathematical foundations of probability, contrasting the Classical (a priori), Relative Frequency (empirical), and Subjective conceptual paradigms. Historically established by Pierre-Simon Laplace, the Classical Definition of Probability states that if a random experiment can result in n mutually exclusive and equally likely elementary outcomes, and if m of these outcomes are favorable to event A, the probability of event A is P(A) = m / n. While computationally straightforward for symmetric games of chance such as dice rolls and coin tosses, the classical approach has notable theoretical limitations: it assumes outcomes are equally likely (creating a circular definition) and cannot be applied when possible outcomes are infinite or asymmetric. To overcome these limitations, the Relative Frequency (Empirical / a posteriori) Definition was developed by Richard von Mises. Under this framework, probability is defined as the limiting relative frequency of an event as the number of experimental repetitions approaches infinity: P(A) = lim [m / n] as n -> inf, where m is the observed frequency of event A across n identical trials. This empirical approach anchors the Law of Large Numbers, proving that sample proportions converge toward true population probabilities as sample size expands. The lecture also introduces Subjective Probability, where probability represents an individual's quantified degree of personal belief regarding an uncertain event, incorporating subjective judgment and qualitative evidence. Regardless of interpretation, all valid probability measures satisfy fundamental numerical boundaries: the probability of any event is non-negative and cannot exceed unity (0 <= P(A) <= 1); the probability of an impossible event is zero (P(phi) = 0); and the probability of the certain event is unity (P(S) = 1). In Virtual University midterm examinations, comparing classical versus relative frequency definitions, identifying limitations, and calculating sample space probabilities are key exam topics. Studying sta301 handouts and solved papers ensures thorough preparation. In preparing for Virtual University assessments in STA301 Statistics and Probability, students should thoroughly review past paper subjective questions and verify key formulas relating to Classical and Relative Frequency Probability.