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 be a probability space, and let
be events for which
for
,
,
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and for all
. Then one has the complete probability formula:
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The complete probability formula also holds when the number of events is infinite.
The complete probability formula holds for mathematical expectations. Let ,
, be a random variable on
, let
be its mathematical expectation and
the conditional mathematical expectations with respect to events
which form a complete set of alternatives. Then
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A complete set of alternatives is also called a partition of the sample space. A collection of events forms a partition if the events are disjoint, have positive probability and if their union is the sample space.
Complete probability formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_probability_formula&oldid=15372