# Complete probability formula

A relationship enabling one to calculate the unconditional probability of an event via its conditional probabilities with respect to events forming a complete set of alternatives.

More precisely, let $( \Omega , {\mathcal A} , {\mathsf P})$ be a probability space, and let $A, A _ {1} \dots A _ {n} \in {\mathcal A}$ be events for which $A _ {i} \cap A _ {j} = \emptyset$ for $i \neq j$, $i, j = 1 \dots n$,

$$\cup _ { k= } 1 ^ { n } A _ {k} = \Omega ,$$

and ${\mathsf P} ( A _ {k} ) > 0$ for all $k$. Then one has the complete probability formula:

$${\mathsf P} ( A) = \sum _ { k= } 1 ^ { n } {\mathsf P} ( A \mid A _ {k} ) {\mathsf P} ( A _ {k} ).$$

The complete probability formula also holds when the number of events $A _ {1} , A _ {2} \dots$ is infinite.

The complete probability formula holds for mathematical expectations. Let $X ( \omega )$, $\omega \in \Omega$, be a random variable on $( \Omega , {\mathcal A} , {\mathsf P})$, let ${\mathsf E} X$ be its mathematical expectation and ${\mathsf E}( X \mid A _ {k} )$ the conditional mathematical expectations with respect to events $A _ {k}$ which form a complete set of alternatives. Then

$${\mathsf E} X = \sum _ { k } {\mathsf E} ( X \mid A _ {k} ) {\mathsf P} ( A _ {k} ).$$

A complete set of alternatives is also called a partition of the sample space. A collection of events $A _ {k}$ forms a partition if the events are disjoint, have positive probability and if their union is the sample space.