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''event''
 
''event''
  
 
Any combination of outcomes of an experiment that has a definite probability of occurrence.
 
Any combination of outcomes of an experiment that has a definite probability of occurrence.
  
Example 1. In the throwing of two dice, each of the 36 outcomes can be represented as a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r0772901.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r0772902.png" /> is the number of dots on the upper face of the first dice and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r0772903.png" /> the number on the second. The event  "the sum of the dots is equal to 11"  is just the combination of the two outcomes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r0772904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r0772905.png" />.
+
Example 1. In the throwing of two dice, each of the 36 outcomes can be represented as a pair $  ( i , j ) $,  
 
+
where $  i $
Example 2. In the random throwing of two points into an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r0772906.png" />, the set of all outcomes can be represented as the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r0772907.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r0772908.png" /> is the value of the first point and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r0772909.png" /> that of the second) in the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729010.png" />. The event "the length of the interval joining x and y is less than a, 0&lt;a&lt; 1" is just the set of points in the square whose distance from the diagonal passing through the origin is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729011.png" />.
+
is the number of dots on the upper face of the first dice and $  j $
 
+
the number on the second. The event  "the sum of the dots is equal to 11"  is just the combination of the two outcomes  $ ( 5 , 6 ) $
Within the limits of the generally accepted axiomatics of [[Probability theory|probability theory]] (see [[#References|[1]]]), where at the base of the probability model lies a [[Probability space|probability space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729013.png" /> is a space of elementary events, i.e. the set of all possible outcomes of a given experiment, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729014.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729015.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729017.png" /> is a probability measure defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729018.png" />), random events are just the sets which belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729019.png" />.
+
and  $ ( 6 , 5 ) $.
 
 
In the first of the above examples, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729020.png" /> is a finite set of 36 elements: the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729022.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729023.png" /> is the class of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729024.png" /> subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729025.png" /> (including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729026.png" /> itself and the empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729027.png" />), and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729028.png" /> the probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729029.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729031.png" /> is the number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729032.png" />. In the second example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729033.png" /> is the set of points in the unit square, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729034.png" /> is the class of its Borel subsets and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729035.png" /> is ordinary Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729036.png" /> (which for simple figures coincides with their area).
 
 
 
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729037.png" /> of events associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729038.png" /> forms a [[Boolean ring|Boolean ring]] with identity with respect to the operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729039.png" /> (symmetric difference) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729040.png" /> (it has a multiplicative identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729041.png" />), that is, it forms a [[Boolean algebra|Boolean algebra]]. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729042.png" /> defined on this Boolean algebra has all the properties of a norm except one: it does not follow from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729043.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729044.png" />. By declaring two events to be equivalent if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729045.png" />-measure of their symmetric difference is zero, and considering equivalence classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729046.png" /> instead of events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729047.png" />, one obtains the normalized Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729048.png" /> of classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077290/r07729049.png" />. This observation leads to another possible approach to the axiomatics of probability theory, in which the basic object is not the probability space connected with a given experiment, but a normalized Boolean algebra of random events (see [[#References|[2]]], [[#References|[3]]]).
 
 
 
====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">  B.V. Gnedenko,   A.N. Kolmogorov,  "Probability theory" , ''Mathematics in the USSR during thirty years: 1917–1947'' , Moscow-Leningrad  (1948)  pp. 701–727  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. Kolmogorov,  "Algèbres de Boole métriques complètes" , ''VI Zjazd Mathematyków Polskich'' , Kraków  (1950)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR></table>
 
  
 +
Example 2. In the random throwing of two points into an interval  $  [ 0 , 1 ] $,
 +
the set of all outcomes can be represented as the set of points  $  ( x , y ) $(
 +
where  $  x $
 +
is the value of the first point and  $  y $
 +
that of the second) in the square  $  \{ {( x , y ) } : {0 \leq  x \leq  1,  0 \leq  y \leq  1 } \} $.
 +
The event  "the length of the interval joining x and y is less than a, 0<a< 1"  is just the set of points in the square whose distance from the diagonal passing through the origin is less than  $  \alpha \sqrt 2 $.
  
 +
Within the limits of the generally accepted axiomatics of [[Probability theory|probability theory]] (see [[#References|[1]]]), where at the base of the probability model lies a [[Probability space|probability space]]  $  ( \Omega , {\mathcal A} , {\mathsf P} ) $(
 +
$  \Omega $
 +
is a space of elementary events, i.e. the set of all possible outcomes of a given experiment,  $  {\mathcal A} $
 +
is a  $  \sigma $-
 +
algebra of subsets of  $  \Omega $
 +
and  $  {\mathsf P} $
 +
is a probability measure defined on  $  {\mathcal A} $),
 +
random events are just the sets which belong to  $  {\mathcal A} $.
  
====Comments====
+
In the first of the above examples,  $  \Omega $
 +
is a finite set of 36 elements: the pairs  $  ( i , j ) $,
 +
$  1 \leq  i , j \leq  6 $;
 +
$  {\mathcal A} $
 +
is the class of all  $  2  ^ {36} $
 +
subsets of  $  \Omega $(
 +
including  $  \Omega $
 +
itself and the empty set  $  \emptyset $),
 +
and for every  $  A \in {\mathcal A} $
 +
the probability  $  {\mathsf P} ( A) $
 +
is equal to  $  m / 36 $,
 +
where  $  m $
 +
is the number of elements of  $  A $.
 +
In the second example,  $  \Omega $
 +
is the set of points in the unit square,  $  {\mathcal A} $
 +
is the class of its Borel subsets and  $  {\mathsf P} $
 +
is ordinary Lebesgue measure on  $  {\mathcal A} $(
 +
which for simple figures coincides with their area).
  
 +
The class  $  {\mathcal A} $
 +
of events associated with  $  ( \Omega , {\mathcal A} , {\mathsf P} ) $
 +
forms a [[Boolean ring|Boolean ring]] with identity with respect to the operations  $  A + B = ( A \setminus  B ) \cup ( B \setminus  A ) $(
 +
symmetric difference) and  $  A \cdot B = A \cap B $(
 +
it has a multiplicative identity  $  \Omega $),
 +
that is, it forms a [[Boolean algebra|Boolean algebra]]. The function  $  {\mathsf P} ( A) $
 +
defined on this Boolean algebra has all the properties of a norm except one: it does not follow from  $  {\mathsf P} ( A) = 0 $
 +
that  $  A = \emptyset $.
 +
By declaring two events to be equivalent if the  $  {\mathsf P} $-
 +
measure of their symmetric difference is zero, and considering equivalence classes  $  \overline{A}\; $
 +
instead of events  $  A $,
 +
one obtains the normalized Boolean algebra  $  \overline {\mathcal A} \; $
 +
of classes  $  \overline{A}\; $.
 +
This observation leads to another possible approach to the axiomatics of probability theory, in which the basic object is not the probability space connected with a given experiment, but a normalized Boolean algebra of random events (see [[#References|[2]]], [[#References|[3]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1''', Wiley (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Bauer, "Probability theory and elements of measure theory", Holt, Rinehart &amp; Winston (1972) pp. Chapt. 11 (Translated from German)</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">  B.V. Gnedenko,  A.N. Kolmogorov,  "Probability theory" , ''Mathematics in the USSR during thirty years: 1917–1947'' , Moscow-Leningrad  (1948)  pp. 701–727  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. Kolmogorov,  "Algèbres de Boole métriques complètes" , ''VI Zjazd Mathematyków Polskich'' , Kraków  (1950)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1''', Wiley (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Bauer, "Probability theory and elements of measure theory", Holt, Rinehart &amp; Winston (1972) pp. Chapt. 11 (Translated from German)</TD></TR></table>

Latest revision as of 07:40, 27 January 2024


event

Any combination of outcomes of an experiment that has a definite probability of occurrence.

Example 1. In the throwing of two dice, each of the 36 outcomes can be represented as a pair $ ( i , j ) $, where $ i $ is the number of dots on the upper face of the first dice and $ j $ the number on the second. The event "the sum of the dots is equal to 11" is just the combination of the two outcomes $ ( 5 , 6 ) $ and $ ( 6 , 5 ) $.

Example 2. In the random throwing of two points into an interval $ [ 0 , 1 ] $, the set of all outcomes can be represented as the set of points $ ( x , y ) $( where $ x $ is the value of the first point and $ y $ that of the second) in the square $ \{ {( x , y ) } : {0 \leq x \leq 1, 0 \leq y \leq 1 } \} $. The event "the length of the interval joining x and y is less than a, 0<a< 1" is just the set of points in the square whose distance from the diagonal passing through the origin is less than $ \alpha \sqrt 2 $.

Within the limits of the generally accepted axiomatics of probability theory (see [1]), where at the base of the probability model lies a probability space $ ( \Omega , {\mathcal A} , {\mathsf P} ) $( $ \Omega $ is a space of elementary events, i.e. the set of all possible outcomes of a given experiment, $ {\mathcal A} $ is a $ \sigma $- algebra of subsets of $ \Omega $ and $ {\mathsf P} $ is a probability measure defined on $ {\mathcal A} $), random events are just the sets which belong to $ {\mathcal A} $.

In the first of the above examples, $ \Omega $ is a finite set of 36 elements: the pairs $ ( i , j ) $, $ 1 \leq i , j \leq 6 $; $ {\mathcal A} $ is the class of all $ 2 ^ {36} $ subsets of $ \Omega $( including $ \Omega $ itself and the empty set $ \emptyset $), and for every $ A \in {\mathcal A} $ the probability $ {\mathsf P} ( A) $ is equal to $ m / 36 $, where $ m $ is the number of elements of $ A $. In the second example, $ \Omega $ is the set of points in the unit square, $ {\mathcal A} $ is the class of its Borel subsets and $ {\mathsf P} $ is ordinary Lebesgue measure on $ {\mathcal A} $( which for simple figures coincides with their area).

The class $ {\mathcal A} $ of events associated with $ ( \Omega , {\mathcal A} , {\mathsf P} ) $ forms a Boolean ring with identity with respect to the operations $ A + B = ( A \setminus B ) \cup ( B \setminus A ) $( symmetric difference) and $ A \cdot B = A \cap B $( it has a multiplicative identity $ \Omega $), that is, it forms a Boolean algebra. The function $ {\mathsf P} ( A) $ defined on this Boolean algebra has all the properties of a norm except one: it does not follow from $ {\mathsf P} ( A) = 0 $ that $ A = \emptyset $. By declaring two events to be equivalent if the $ {\mathsf P} $- measure of their symmetric difference is zero, and considering equivalence classes $ \overline{A}\; $ instead of events $ A $, one obtains the normalized Boolean algebra $ \overline {\mathcal A} \; $ of classes $ \overline{A}\; $. This observation leads to another possible approach to the axiomatics of probability theory, in which the basic object is not the probability space connected with a given experiment, but a normalized Boolean algebra of random events (see [2], [3]).

References

[1] A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)
[2] B.V. Gnedenko, A.N. Kolmogorov, "Probability theory" , Mathematics in the USSR during thirty years: 1917–1947 , Moscow-Leningrad (1948) pp. 701–727 (In Russian)
[3] A.N. Kolmogorov, "Algèbres de Boole métriques complètes" , VI Zjazd Mathematyków Polskich , Kraków (1950)
[4] P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[a1] W. Feller, "An introduction to probability theory and its applications", 1, Wiley (1957)
[a2] H. Bauer, "Probability theory and elements of measure theory", Holt, Rinehart & Winston (1972) pp. Chapt. 11 (Translated from German)
How to Cite This Entry:
Random event. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Random_event&oldid=25926
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article