In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events, hence the name.

The law of total probability is a theorem that states, in its discrete case, if ${\displaystyle \left\{{B_{n}:n=1,2,3,\ldots }\right\}}$ is a finite or countably infinite set of mutually exclusive and collectively exhaustive events, then for any event ${\displaystyle A}$

$${\displaystyle P(A)=\sum _{n}P(A\cap B_{n})}$$

or, alternatively,

$${\displaystyle P(A)=\sum _{n}P(A\mid B_{n})P(B_{n}),}$$

where, for any ${\displaystyle n}$, if ${\displaystyle P(B_{n})=0}$, then these terms are simply omitted from the summation since ${\displaystyle P(A\mid B_{n})}$ is finite.

The summation can be interpreted as a weighted average, and consequently the marginal probability, ${\displaystyle P(A)}$, is sometimes called "average probability"; "overall probability" is sometimes used in less formal writings.

The law of total probability can also be stated for conditional probabilities: