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The conditional probability of an event relative to another event is a characteristic connecting the two events. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c0245101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c0245102.png" /> are events and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c0245103.png" />, then the conditional probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c0245104.png" /> of the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c0245105.png" /> relative to (or under the condition, or with respect to) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c0245106.png" /> is defined by the equation
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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/c024/c024510/c0245107.png" /></td> </tr></table>
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The conditional probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c0245108.png" /> can be regarded as the probability that the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c0245109.png" /> is realized under the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451010.png" /> has taken place. For independent events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451012.png" /> the conditional probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451013.png" /> coincides with the unconditional probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451014.png" />.
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The conditional probability of an event relative to another event is a characteristic connecting the two events. If  $  A $
 +
and  $  B $
 +
are events and  $  {\mathsf P} ( B) > 0 $,
 +
then the conditional probability  $  {\mathsf P} ( A \mid  B  ) $
 +
of the event  $  A $
 +
relative to (or under the condition, or with respect to)  $  B $
 +
is defined by the equation
 +
 
 +
$$
 +
{\mathsf P} ( A \mid  B  )  = \
 +
 
 +
\frac{ {\mathsf P} ( A \cap B  ) }{ {\mathsf P} ( B  ) }
 +
.
 +
$$
 +
 
 +
The conditional probability  $  {\mathsf P} ( A \mid  B  ) $
 +
can be regarded as the probability that the event $  A $
 +
is realized under the condition that $  B $
 +
has taken place. For independent events $  A $
 +
and $  B $
 +
the conditional probability $  {\mathsf P} ( A \mid  B  ) $
 +
coincides with the unconditional probability $  {\mathsf P} ( A) $.
  
 
About the connection between the conditional and unconditional probabilities of events see [[Bayes formula|Bayes formula]] and [[Complete probability formula|Complete probability formula]].
 
About the connection between the conditional and unconditional probabilities of events see [[Bayes formula|Bayes formula]] and [[Complete probability formula|Complete probability formula]].
  
The conditional probability of an event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451016.png" /> with respect to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451017.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451018.png" /> is a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451019.png" />, measurable relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451020.png" />, for which
+
The conditional probability of an event $  A $
 +
with respect to a $  \sigma $-
 +
algebra $  \mathfrak B $
 +
is a random variable $  {\mathsf P} ( A \mid  \mathfrak B  ) $,  
 +
measurable relative to $  \mathfrak B $,  
 +
for which
  
<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/c024/c024510/c02451021.png" /></td> </tr></table>
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$$
 +
\int\limits _ { B }
 +
{\mathsf P} ( A \mid  \mathfrak B  )
 +
{\mathsf P} ( d \omega )  = \
 +
{\mathsf P} ( A \cap B  )
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451022.png" />. The conditional probability with respect to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451023.png" />-algebra is defined up to equivalence.
+
for any $  B \in \mathfrak B $.  
 +
The conditional probability with respect to a $  \sigma $-
 +
algebra is defined up to equivalence.
  
If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451024.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451025.png" /> is generated by a countable number of disjoint events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451026.png" /> having positive probability and the union of which coincides with the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451027.png" />, then
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If the $  \sigma $-
 +
algebra $  \mathfrak B $
 +
is generated by a countable number of disjoint events $  B _ {1} , B _ {2} \dots $
 +
having positive probability and the union of which coincides with the whole space $  \Omega $,  
 +
then
  
<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/c024/c024510/c02451028.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} ( A \mid  \mathfrak B  )  = \
 +
{\mathsf P} ( A \mid  B _ {k}  ) \ \
 +
\textrm{ for }  \omega \in B _ {k} ,\ \
 +
k = 1 , 2 ,\dots .
 +
$$
  
The conditional probability of an event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451029.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451030.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451031.png" /> can be defined as the [[Conditional mathematical expectation|conditional mathematical expectation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451032.png" /> of the indicator function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451033.png" />.
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The conditional probability of an event $  A $
 +
with respect to the $  \sigma $-
 +
algebra $  \mathfrak B $
 +
can be defined as the [[Conditional mathematical expectation|conditional mathematical expectation]] $  {\mathsf E} ( I _ {A} \mid  \mathfrak B  ) $
 +
of the indicator function of $  A $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451034.png" /> be a probability space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451035.png" /> be a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451036.png" />. The conditional probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451037.png" /> is called regular if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451040.png" />, such that
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Let $  ( \Omega , {\mathcal A} , {\mathsf P}  ) $
 +
be a probability space and let $  \mathfrak B $
 +
be a subalgebra of $  {\mathcal A} $.  
 +
The conditional probability $  {\mathsf P} ( A \mid  \mathfrak B  ) $
 +
is called regular if there exists a function $  p ( \omega , A ) $,  
 +
$  \omega \in \Omega $,  
 +
$  A \in {\mathcal A} $,  
 +
such that
  
a) for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451041.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451042.png" /> is a probability on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451043.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451044.png" />;
+
a) for a fixed $  \omega $
 +
the function $  p ( \omega , A ) $
 +
is a probability on the $  \sigma $-
 +
algebra $  {\mathcal A} $;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451045.png" /> with probability one.
+
b) $  {\mathsf P} ( A \mid  \mathfrak B  ) = p ( \omega , A ) $
 +
with probability one.
  
 
For a regular conditional probability the conditional mathematical expectation can be expressed by integrals, with the conditional probability taking the role of the measure.
 
For a regular conditional probability the conditional mathematical expectation can be expressed by integrals, with the conditional probability taking the role of the measure.
  
The conditional probability with respect to a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451046.png" /> is defined as the conditional probability with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451047.png" />-algebra generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024510/c02451048.png" />.
+
The conditional probability with respect to a random variable $  X $
 +
is defined as the conditional probability with respect to the $  \sigma $-
 +
algebra generated by $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  "Foundations of the theory of probability" , Chelsea, reprint  (1950)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Loève,  "Probability theory" , Princeton Univ. Press  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  "Foundations of the theory of probability" , Chelsea, reprint  (1950)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Loève,  "Probability theory" , Princeton Univ. Press  (1963)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


The conditional probability of an event relative to another event is a characteristic connecting the two events. If $ A $ and $ B $ are events and $ {\mathsf P} ( B) > 0 $, then the conditional probability $ {\mathsf P} ( A \mid B ) $ of the event $ A $ relative to (or under the condition, or with respect to) $ B $ is defined by the equation

$$ {\mathsf P} ( A \mid B ) = \ \frac{ {\mathsf P} ( A \cap B ) }{ {\mathsf P} ( B ) } . $$

The conditional probability $ {\mathsf P} ( A \mid B ) $ can be regarded as the probability that the event $ A $ is realized under the condition that $ B $ has taken place. For independent events $ A $ and $ B $ the conditional probability $ {\mathsf P} ( A \mid B ) $ coincides with the unconditional probability $ {\mathsf P} ( A) $.

About the connection between the conditional and unconditional probabilities of events see Bayes formula and Complete probability formula.

The conditional probability of an event $ A $ with respect to a $ \sigma $- algebra $ \mathfrak B $ is a random variable $ {\mathsf P} ( A \mid \mathfrak B ) $, measurable relative to $ \mathfrak B $, for which

$$ \int\limits _ { B } {\mathsf P} ( A \mid \mathfrak B ) {\mathsf P} ( d \omega ) = \ {\mathsf P} ( A \cap B ) $$

for any $ B \in \mathfrak B $. The conditional probability with respect to a $ \sigma $- algebra is defined up to equivalence.

If the $ \sigma $- algebra $ \mathfrak B $ is generated by a countable number of disjoint events $ B _ {1} , B _ {2} \dots $ having positive probability and the union of which coincides with the whole space $ \Omega $, then

$$ {\mathsf P} ( A \mid \mathfrak B ) = \ {\mathsf P} ( A \mid B _ {k} ) \ \ \textrm{ for } \omega \in B _ {k} ,\ \ k = 1 , 2 ,\dots . $$

The conditional probability of an event $ A $ with respect to the $ \sigma $- algebra $ \mathfrak B $ can be defined as the conditional mathematical expectation $ {\mathsf E} ( I _ {A} \mid \mathfrak B ) $ of the indicator function of $ A $.

Let $ ( \Omega , {\mathcal A} , {\mathsf P} ) $ be a probability space and let $ \mathfrak B $ be a subalgebra of $ {\mathcal A} $. The conditional probability $ {\mathsf P} ( A \mid \mathfrak B ) $ is called regular if there exists a function $ p ( \omega , A ) $, $ \omega \in \Omega $, $ A \in {\mathcal A} $, such that

a) for a fixed $ \omega $ the function $ p ( \omega , A ) $ is a probability on the $ \sigma $- algebra $ {\mathcal A} $;

b) $ {\mathsf P} ( A \mid \mathfrak B ) = p ( \omega , A ) $ with probability one.

For a regular conditional probability the conditional mathematical expectation can be expressed by integrals, with the conditional probability taking the role of the measure.

The conditional probability with respect to a random variable $ X $ is defined as the conditional probability with respect to the $ \sigma $- algebra generated by $ X $.

References

[1] A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)
[2] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
[3] M. Loève, "Probability theory" , Princeton Univ. Press (1963)
How to Cite This Entry:
Conditional probability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_probability&oldid=15106
This article was adapted from an original article by V.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article