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Difference between revisions of "Complete probability formula"

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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.
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More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c0238301.png" /> be a probability space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c0238302.png" /> be events for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c0238303.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c0238304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c0238305.png" />,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c0238306.png" /></td> </tr></table>
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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.
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c0238307.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c0238308.png" />. Then one has the complete probability formula:
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More precisely, let  $  ( \Omega , {\mathcal A} , {\mathsf P}) $
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be a probability space, and let  $  A, A _ {1} \dots A _ {n} \in {\mathcal A} $
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be events for which  $  A _ {i} \cap A _ {j} = \emptyset $
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for $  i \neq j $,
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$  i, j = 1 \dots n $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c0238309.png" /></td> </tr></table>
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$$
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\cup _ { k= } 1 ^ { n }  A _ {k}  = \Omega ,
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$$
  
The complete probability formula also holds when the number of events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c02383010.png" /> is infinite.
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and  $  {\mathsf P} ( A _ {k} ) > 0 $
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for all  $  k $.
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Then one has the complete probability formula:
  
The complete probability formula holds for mathematical expectations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c02383011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c02383012.png" />, be a random variable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c02383013.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c02383014.png" /> be its mathematical expectation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c02383015.png" /> the conditional mathematical expectations with respect to events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c02383016.png" /> which form a complete set of alternatives. Then
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$$
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{\mathsf P} ( A)  = \sum _ { k= } 1 ^ { n }  {\mathsf P} ( A \mid  A _ {k} ) {\mathsf P} ( A _ {k} ).
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c02383017.png" /></td> </tr></table>
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The complete probability formula also holds when the number of events  $  A _ {1} , A _ {2} \dots $
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is infinite.
  
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The complete probability formula holds for mathematical expectations. Let  $  X ( \omega ) $,
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$  \omega \in \Omega $,
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be a random variable on  $  ( \Omega , {\mathcal A} , {\mathsf P}) $,
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let  $  {\mathsf E} X $
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be its mathematical expectation and  $  {\mathsf E}( X \mid  A _ {k} ) $
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the conditional mathematical expectations with respect to events  $  A _ {k} $
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which form a complete set of alternatives. Then
  
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$$
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{\mathsf E} X  =  \sum _ { k } {\mathsf E} ( X \mid  A _ {k} ) {\mathsf P} ( A _ {k} ).
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$$
  
 
====Comments====
 
====Comments====
A complete set of alternatives is also called a partition of the sample space. A collection of events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023830/c02383018.png" /> forms a partition if the events are disjoint, have positive probability and if their union is the sample space.
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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.

Revision as of 17:45, 4 June 2020


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.

How to Cite This Entry:
Complete probability formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_probability_formula&oldid=15372
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article