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Difference between revisions of "Certain event"

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An event which is known a priori to be certain to occur. More exactly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021330/c0213301.png" /> is the space of elementary results, an event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021330/c0213302.png" /> which occurs together with any of the elementary results <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021330/c0213303.png" /> is said to be certain, and must clearly coincide with the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021330/c0213304.png" />. It is then natural to ascribe the probability 1 to a certain event:
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An [[Impossible event|impossible event]] is an event complementary to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021330/c0213306.png" />.
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An event which is known a priori to be certain to occur. More exactly, if  $  \Omega = \{ \omega \} $
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is the space of elementary results, an event  $  A $
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which occurs together with any of the elementary results  $  \omega $
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is said to be certain, and must clearly coincide with the entire space  $  \Omega $.
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It is then natural to ascribe the probability 1 to a certain event:
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$$
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{\mathsf P} ( A)  =  \textrm{ measure } \{  \omega  : {\omega \in A } \}
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=  {\mathsf P} ( \Omega )
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=  1 .
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$$
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An [[Impossible event|impossible event]] is an event complementary to $  A $.

Latest revision as of 16:43, 4 June 2020


An event which is known a priori to be certain to occur. More exactly, if $ \Omega = \{ \omega \} $ is the space of elementary results, an event $ A $ which occurs together with any of the elementary results $ \omega $ is said to be certain, and must clearly coincide with the entire space $ \Omega $. It is then natural to ascribe the probability 1 to a certain event:

$$ {\mathsf P} ( A) = \textrm{ measure } \{ \omega : {\omega \in A } \} = {\mathsf P} ( \Omega ) = 1 . $$

An impossible event is an event complementary to $ A $.

How to Cite This Entry:
Certain event. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Certain_event&oldid=16962
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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