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

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

Comments

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.