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484:) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event consisting of only a single outcome is called an
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of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the 𝜎-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all
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with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any
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Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard
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1712:. Graduate texts in mathematics. Translated by Boas, Ralph Philip; Chibisov, Dmitry (3rd ed.). New York Heidelberg Dordrecht London: Springer.
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1067:. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as
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1032:{\displaystyle \mathrm {P} (A)={\frac {|A|}{|\Omega |}}\,\ \left({\text{alternatively:}}\ \Pr(A)={\frac {|A|}{|\Omega |}}\right)}
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Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Ludolf Erwin, Meester (2005). Dekking, Michel (ed.).
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of the sample space are defined as events). However, this approach does not work well in cases where the sample space is
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Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using
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it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see
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of the sample space that contain multiple elements. So, for example, potential events include:
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In statistics and probability theory, set of outcomes to which a probability is assigned
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is finite, any subset of the sample space is an event (that is, all elements of the
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A modern introduction to probability and statistics: understanding why and how
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Algebra and trigonometry: Functions and
Applications, Teacher's edition
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Probability, statistics and random processes for electrical engineering
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This rule can readily be applied to each of the example events above.
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1055:, the sample space is the set of real numbers or some subset of the
828:"Red and black at the same time without being a joker" (0 elements),
1242:{\displaystyle \{\omega \in \Omega \mid u<X(\omega )\leq v\}\,}
1574: – Set of random variables of which any two are independent
1687:. Springer texts in statistics. London : Springer. p. 14.
498:. An event that has more than one possible outcome is called a
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is a real-valued random variable defined on the sample space
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they are often written as predicates or indicators involving
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Pages displaying short descriptions of redirect targets
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Even though events are subsets of some sample space
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1357:{\displaystyle \Pr(u<X\leq v)=F(v)-F(u)\,.}
1652:(Classics ed.). Upper Saddle River, NJ:
1249:can be written more conveniently as, simply,
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1285:This is especially common in formulas for a
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120:Learn how and when to remove this message
1476:{\displaystyle \omega \in X^{-1}((u,v])}
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694:occurring), and together these define a
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1544: – Opposite of a probability event
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1520:{\displaystyle u<X(\omega )\leq v.}
1091:sets proves more useful in practice.
7:
1075:, to work, it is necessary to use a
58:adding citations to reliable sources
1598:. Upper Saddle River, NJ: Pearson.
719:§ Events in probability spaces
1625:. Dover Publications. p. 18.
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1111:are elements of the 𝜎-algebra.
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1775:Experiment (probability theory)
1278:{\displaystyle u<X\leq v\,.}
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831:"The 5 of Hearts" (1 element),
698:: did the event occur or not?
217:Collectively exhaustive events
1:
1592:Leon-Garcia, Alberto (2008).
1063:sets, such as those that are
1708:Širjaev, Alʹbert N. (2016).
1392:{\displaystyle u<X\leq v}
1043:Events in probability spaces
837:"A Face card" (12 elements),
729:If we assemble a deck of 52
1743:Encyclopedia of Mathematics
1572:Pairwise independent events
1791:
1646:Foerster, Paul A. (2006).
1619:Pfeiffer, Paul E. (1978).
1073:conditional probabilities
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1182:{\displaystyle \Omega ,}
1135:{\displaystyle \Omega ,}
840:"A Spade" (13 elements),
781:is the sample space and
570:(or trial) (that is, if
387:Law of total probability
382:Conditional independence
271:Exponential distribution
256:Probability distribution
846:"A card" (52 elements).
686:). An event defines a
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392:Law of large numbers
361:Marginal probability
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135:Part of a series on
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1542:Complementary event
1089:Lebesgue measurable
1053:normal distribution
688:complementary event
598:probability measure
494:; that is, it is a
351:Complementary event
293:Probability measure
281:Pareto distribution
276:Normal distribution
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1146:. For example, if
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227:Mutual exclusivity
144:Probability theory
1755:Formal definition
1719:978-0-387-72205-4
1694:978-1-85233-896-1
1632:978-0-486-63677-1
1557:Independent event
1419:{\displaystyle X}
1159:{\displaystyle X}
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867:{\displaystyle P}
816:{\displaystyle A}
794:{\displaystyle A}
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715:probability space
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633:{\displaystyle S}
613:{\displaystyle S}
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539:{\displaystyle S}
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186:Indeterminism
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71: –
70:
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65:Find sources:
59:
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43:This article
41:
37:
32:
31:
19:
1759:Mizar system
1741:
1709:
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1594:
1587:
1118:
1093:
1057:real numbers
1046:
874:of an event
849:
801:is an event.
728:
703:sample space
700:
499:
492:atomic event
482:sample space
461:
455:
418:Tree diagram
413:Venn diagram
377:Independence
323:Markov chain
211:
207:Sample space
116:
110:January 2018
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
18:Random event
1287:probability
522:is said to
333:Random walk
174:Determinism
162:Probability
1656:. p.
1403:under the
1289:, such as
1189:the event
1104:𝜎-algebra
721:, below).
568:experiment
474:experiment
244:Experiment
191:Randomness
137:statistics
80:newspapers
1748:EMS Press
1509:≤
1503:ω
1445:−
1437:∈
1434:ω
1384:≤
1336:−
1312:≤
1266:≤
1230:≤
1224:ω
1209:∣
1206:Ω
1203:∈
1200:ω
1174:Ω
1127:Ω
1014:Ω
948:Ω
744:empty set
707:power set
671:∈
581:∈
502:An event
237:Singleton
1769:Category
1530:See also
1426:because
470:outcomes
318:Variance
1757:in the
1750:, 2001
1405:mapping
1083:is the
897:formula
742:), the
566:of the
480:of the
232:Outcome
94:scholar
1716:
1691:
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1629:
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973:
960:
490:or an
478:subset
472:of an
179:System
167:Axioms
96:
89:
82:
75:
67:
1579:Notes
1069:joint
524:occur
464:is a
462:event
460:, an
212:Event
101:JSTOR
87:books
1714:ISBN
1689:ISBN
1662:ISBN
1627:ISBN
1600:ISBN
1494:<
1378:<
1364:The
1306:<
1260:<
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1071:and
738:(an
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1658:634
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757:An
692:not
526:if
476:(a
468:of
466:set
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445:e
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