Law of Total Probability

The probability of an event equals the weighted average of its conditional probabilities across a partition of the sample space.

Partitioning the sample space

A partition is a collection of disjoint events B1, …, Bn whose union is the whole sample space S. Every outcome lands in exactly one Bi. Partitions let you break a hard probability into cases you can handle.

The law

For any event A and any partition {Bi}, P(A) = Σ P(A | Bi) P(Bi). Each term weights the conditional probability of A within case Bi by how likely that case is. The result is a probability-weighted average.

A simple case

The two-case version is P(A) = P(A | B) P(B) + P(A | Bᶜ) P(Bᶜ). This is the denominator that appears in Bayes theorem, which is why the two topics are always taught together.

Worked example

Three suppliers provide 50%, 30%, and 20% of a part, with defect rates 1%, 2%, and 3%. The overall defect rate is 0.5·0.01 + 0.3·0.02 + 0.2·0.03 = 0.017, or 1.7%. No single supplier has that rate; it is the mixture.

Why it is useful

The law converts an unconditional question you cannot answer directly into a set of conditional questions you can. In Monte Carlo estimation it justifies stratified sampling: split the input space into regions, estimate within each, then recombine with the correct weights. Done well, stratification lowers variance for the same number of samples.

The continuous analogue replaces the sum with an integral over a conditioning variable, P(A) = ∫ P(A | x) f(x) dx.