Law of total probability

In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities.

Statement

The law of total probability is the proposition that if $\left\{{B_{n}:n=1,2,3,\ldots }\right\}$ is a finite or countably infinite partition of a sample space (in other words, a set of pairwise disjoint events whose union is the entire sample space) and each event $B_{n}$ is measurable, then for any event $A$ of the same probability space:

$\Pr(A)=\sum _{n}\Pr(A\cap B_{n})\,$

or, alternatively,

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

where, for any $n\,$ for which $\Pr(B_{n})=0\,$ these terms are simply omitted from the summation, because $\Pr(A\mid B_{n})\,$ is finite.

The summation can be interpreted as a weighted average, and consequently the marginal probability, $\Pr(A)$, 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. Taking the $B_{n}$ as above, and assuming $X$ is not mutually exclusive with $A$ or any of the $B_{n}$:

$\Pr(A\mid X)=\sum _{n}\Pr(A\mid X\cap B_{n})\Pr(B_{n}\mid X)\,$

Applications

One common application of the law is where the events coincide with a discrete random variable X taking each value in its range, i.e. $B_{n}$ is the event $X=x_{n}$. It follows that the probability of an event A is equal to the expected value of the conditional probabilities of A given $X=x_{n}$. That is,

$\Pr(A)=\sum _{n}\Pr(A\mid X=x_{n})\Pr(X=x_{n})=\operatorname {E} _{X}[\Pr(A\mid X)],$

where Pr(A|X) is the conditional probability of A given X, and where E_X denotes the expectation with respect to the random variable X.

This result can be generalized to continuous random variables (via continuous conditional density), and the expression becomes

$\Pr(A)=\operatorname {E} [\Pr(A\mid {\mathcal {F}}_{X})],$

where ${\mathcal {F}}_{X}$ denotes the sigma-algebra generated by the random variable X.

Other names

The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. One author even uses the terminology "continuous law of alternatives" in the continuous case. This result is given by Grimmett and Welsh as the partition theorem, a name that they also give to the related law of total expectation.

See also

• Law of total expectation
• Law of total variance
• Law of total cumulance

References

• Introduction to Probability and Statistics by William Mendenhall, Robert J. Beaver, Barbara M. Beaver, Thomson Brooks/Cole, 2005, page 159.
• Theory of Statistics, by Mark J. Schervish, Springer, 1995.
• Schaum's Outline of Theory and Problems of Beginning Finite Mathematics, by John J. Schiller, Seymour Lipschutz, and R. Alu Srinivasan, McGraw-Hill Professional, 2005, page 116.
• A First Course in Stochastic Models, by H. C. Tijms, John Wiley and Sons, 2003, pages 431–432.
• An Intermediate Course in Probability, by Alan Gut, Springer, 1995, pages 5–6.