Bayes' Theorem and Discrete Random Variables

Lesson 22 of STA301 marks the culmination of the Virtual University midterm syllabus (Lectures 1 to 22), presenting Bayes' Theorem and introducing Random Variables. Formulated by Reverend Thomas Bayes, Bayes' Theorem provides an inferential mechanism for revising initial prior probabilities in light of new observed sample evidence to obtain updated posterior probabilities. The mathematical derivation begins with the Law of Total Probability. If events B1, B2, ... Bk form a partition of the sample space S (meaning they are mutually exclusive and exhaustive: sum B_i = S and B_i cap B_j = phi for i != j), and if A is any arbitrary event in S, then A can be partitioned into disjoint subsets: A = (A cap B1) U (A cap B2) U ... U (A cap Bk). Consequently, the total probability of event A is given by P(A) = sum [P(B_i) * P(A | B_i)]. Bayes' Theorem combines conditional probability with the law of total probability to calculate the posterior probability of a specific partition state B_r given that event A has occurred: P(B_r | A) = [P(B_r) * P(A | B_r)] / sum [P(B_i) * P(A | B_i)]. The lecture illustrates practical applications, including medical diagnostic tests (evaluating the probability of disease given a positive test result) and industrial quality control (identifying machine defect sources). The second portion introduces the fundamental concept of a Random Variable—a real-valued mathematical function X: S -> R that assigns a unique numerical value to every sample point in the sample space. A Discrete Random Variable takes an isolated countable set of values. Its probability behavior is characterized by the Probability Mass Function (PMF) f(x) = P(X = x), which must satisfy two essential conditions: f(x) >= 0 for all values of x, and the sum of all probabilities over the entire support equals unity, sum f(x) = 1. In Virtual University midterm examinations, numerical problems on Bayes' Theorem and PMF condition verifications appear consistently. Mastering this lecture alongside sta301 handouts, short notes 1 to 22, and solved past papers by Moaaz guarantees top exam performance.