Maths: Bayes' Theorem — Finding the Faulty Machine

Bayes’ Theorem is the mathematical engine for updating a belief when new evidence arrives. In this classic factory scenario, you know how often each machine produces components and how often each machine makes a defective one—but the customer only sees the final, mixed pile. The real question is reversed: given that a component is defective, what is the probability it came from a specific machine? That reversal is exactly what conditional probability and Bayes’ Theorem handle. The key is to connect two layers of information. First, the total probability of a defect, P(D), is found by weighting each machine’s defect rate by its production share—this is the law of total probability. Then, Bayes’ Theorem swaps the condition: P(C | D) = [P(D | C) × P(C)] / P(D). The numerator isolates the “path” through machine C, while the denominator sums all possible paths. By comparing these posterior probabilities for A, B, and C, you can see how the initial production shares and defect rates combine to shift the odds—revealing which machine is most likely responsible for a given defect.