Law of Total Probability

The Law of Total Probability says: if $B_1, B_2, \ldots, B_n$ partition the sample space (mutually exclusive, collectively exhaustive), then for any event $A$: $$P(A) = \sum_{i=1}^n P(A \mid B_i)\, P(B_i)$$ Intuition: You cannot compute $P(A)$ directly, but you can compute it within each scenario $B_i$. Weight each conditional probability by how likely that scenario is, and sum. Think of it as a weighted average of probabilities across cases. Concrete example: A factory has two machines. Machine 1 produces 60% of items with a 2% defect rate. Machine 2 produces 40% with a 5% defect rate. What is the overall defect rate? $$P(\text{defect}) = P(\text{defect} \mid M_1)P(M_1) + P(\text{defect} \mid M_2)P(M_2) = 0.02(0.6) + 0.05(0.4) = 0.032$$ The continuous version replaces the sum with an integral: $P(A) = \int P(A \mid X=x)\, f_X(x)\, dx$. When to use: This is the workhorse for computing unconditional probabilities when the problem naturally breaks into cases — different coin types, different urn compositions, different states of the world. It is also the denominator in Bayes' theorem, so mastering it is essential for Bayesian problems. Law of Total Expectation: The same idea extends to expectations: $E[X] = \sum_i E[X \mid B_i]\, P(B_i)$, or in continuous form, $E[X] = E[E[X \mid Y]]$. This "tower property" is fundamental in stochastic processes.