The law of total probability decomposes the probability of an event B into a sum of joint probabilities over a partition of the sample space (disjoint subsets whose union is the entire space), giving P(B) = Σ P(B|Aᵢ)P(Aᵢ), a direct consequence of the additivity axiom of probability applied to conditional probability's multiplication rule. Bayes' rule, P(A|B) = P(B|A)P(A)/P(B), inverts a conditional probability by relating it to the conditional probability in the reverse direction, and is commonly used together with the law of total probability (to compute the denominator) to update a prior probability into a posterior probability given evidence; central related concepts include conditional independence (independence of two events given a third), which is logically distinct from unconditional independence in both directions, and pitfalls such as conflating P(A|B) with P(B|A) or conflating a prior with a posterior probability.
The law of total probability decomposes the probability of an event B into a sum of joint probabilities over a partition of the sample space (disjoint subsets whose union is the entire space), giving…