Namespaces
Variants
Actions

Random event

From Encyclopedia of Mathematics
Jump to: navigation, search


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)

Comments

References

[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=48423
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article