Sample space - Revision history

109.150.118.240: remove two commercial links

2014-12-03T09:19:11Z

remove two commercial links

← Older revision		Revision as of 11:19, 3 December 2014
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	~~{{refimprove\|date=February 2012}}~~		Timika Berumen is the title my mothers and fathers gave me and I completely dig that title. Maryland is where our home is but my spouse wants us to move. My buddies say it's not great for me but what I love performing is taking part in croquet and now I have time to consider on new things. The occupation I've been occupying for many years is a dispatcher. Check out the newest news on her website: http://www.youtube.com/watch?v=nGeC8fsM1N4<br><br>My homepage; [http://www.youtube.com/watch?v=nGeC8fsM1N4 Unlock Her Legs]

	~~{{Probability fundamentals}}~~
	~~In [[probability theory]], an '''event'''~~ is ~~a [[Set (mathematics)\|set]] of [[Outcome (probability)\|outcomes]] of an [[Experiment (probability theory)\|experiment]] (a [[subset]] of~~ the ~~[[sample space]]) to which a probability is assigned.<ref>{{cite book \| last = Leon-Garcia \| first = Alberto \|~~ title ~~= Probability, Statistics~~ and Random Processes for Electrical Engineering \| location = Upper Saddle River, NJ \| publisher = Pearson \| year = 2008 \| url = http://books.google.com/books/about/Probability_Statistics_and_Random_Proces.html?id=GUJosCkbBywC}}</ref> A single outcome may be an element of many different events,<ref>{{cite book \| last = Pfeiffer \| first = Paul E. \| year = 1978 \| title = Concepts of probability theory \| page = 18 \| url = http://books.google.com/books?id=_mayRBczVRwC&pg=PA18 \| publisher = Dover Publications \| ISBN = 978-0-486-63677-1}}</ref> and ~~different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.<ref>{{cite book \| last = Foerster \| first = Paul A. \| year = 2006 \|~~ title = Algebra and Trigonometry: Functions and Applications, Teacher's Edition \| edition = Classics \| page = 634 \| publisher = [[Prentice Hall]] \| location = Upper Saddle River, NJ \| isbn = 0-13-165711-9 \| url = http://www.amazon.~~com/Algebra-Trigonometry-Functions-Applications-Prentice/dp/0131657100}}</ref>~~

	~~Typically, when the sample space is finite, any subset of the sample space~~ is ~~an event (''i''.''e''. all elements of the [[power set]] of the sample space are defined as events). However, this approach does not work well in cases~~ where ~~the sample space~~ is ~~[[uncountably infinite]], most notably when the outcome is a [[real numbers\|real number]]~~. ~~So, when defining a [[probability space]]~~ it ~~is possible, and often necessary, to exclude certain subsets of the sample space from being events (see ''[[#Events in probability spaces\|Events in probability spaces]]'~~'~~, below).~~

	~~==A simple example==~~
	~~If we assemble a deck of 52 [[playing card]]~~s 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 [[singleton set]] (an [[elementary event]]), the [[empty set]] (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are [[proper subset]]s of the sample space that contain multiple elements. So, for ~~example, potential events include:~~
	~~[[Image:Venn A subset B.svg\|thumb\|150px\|A [[Venn diagram]] of an event. ''B''~~ is ~~the sample space~~ and ~~''A'' is an event~~.~~<br>By the ratio of their areas, the probability of ''A'' is approximately 0.4.]]~~

	* "Red and black at the same time without being a joker" (0 elements),
	* "The ~~5 of Hearts" (1 element),~~
	* "A King" (4 elements),
	* "A Face card" (12 elements),
	* "A Spade" (13 elements),
	* "A Face card or a red suit" (32 elements),
	* "A card" (52 elements).
	Since all events are sets, they are usually written as sets (e.g. {1, 2, 3}), and represented graphically using [[Venn diagram]]s. Given that each outcome in the sample space Ω is equally likely, the probability of an event '~~'A'' is the following {{visible anchor\|formula}}: <br />~~
	~~:<math>\mathrm{P}(A) = \frac{\|A\|}{\|\Omega\|}\,\ ( \text{alternatively:}\ \Pr(A) = \frac{\|A\|}{\|\Omega\|})</math>~~
	~~This rule can readily be applied to each of the example events above.~~

	~~==Events in probability spaces==<!--Section linked from lead-->~~
	~~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 [[probability distributions]], such as the [[normal distribution]], the sample space is the set of real numbers or some subset of the [[real numbers]]. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers [[Pathological (mathematics)\|'badly behaved']] sets, such as those that are [[nonmeasurable set\|nonmeasurable]]. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as [[joint probability\|joint]] and [[conditional probability\|conditional probabilities]], to work, it is necessary to use a [[sigma-algebra\|σ-algebra]], that is, a family closed under complementation and countable unions of its members. The most natural choice is the [[Borel measure\|Borel measurable]] set derived from unions and intersections of intervals. However, the larger class of [[Lebesgue measure\|Lebesgue measurable]] sets proves more useful in practice.

	In the general [[measure theory\|measure-theoretic]] description of [[probability space]]s, an event may be defined as an element of a selected [[sigma-algebra\|σ-algebra]] 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 ''events of interest'' are elements of~~ the ~~σ-algebra.~~

	~~==A note~~ on ~~notation==~~
	Even though events are subsets of some sample space Ω, they are often written as [[propositional formula]]s involving [[random variable]]s. For example, if ''X'' is a real-valued random variable defined on the sample space Ω, the event
	:~~<math>\{\omega\in\Omega \mid u < X(\omega) \leq v\}\,<~~/~~math>~~
	~~can be written more conveniently as, simply,~~
	~~:<math>u < X \leq v\,~~.</~~math>~~
	~~This is especially common in formulas for a [[probability]], such as~~
	~~:<math>\Pr(u < X \leq~~ v) = ~~F(v)-F(u)\,.~~<~~/math~~>
	~~The [[set (mathematics)\|set]] ''u''~~ < ~~''X'' ≤ ''v'' is an example of an [[inverse image]] under the [[map (mathematics)\|mapping]] ''X'' because <math~~>~~\omega \in X^{-1}((u, v])</math> if and only if <math>u < X(\omega) \leq v</math>.~~

	~~==See also==~~
	*[[Complementary event]]
	*[[Elementary event]]

	~~==Notes==~~
	~~{{Reflist}}~~

	~~==External links==~~
	*{{springer\|title=Random event\|id=p/r077290}}
	*[http://~~mws~~.cs.~~ru.nl~~/~~mwiki/prob_1.html#M1 Formal definition] in the [[Mizar system]].~~

	~~{{DEFAULTSORT:Event (Probability Theory)}}~~
	~~[[Category:Probability theory]~~]

en>2rock: Added reference to additional explanation of multiple sample spaces, specifically the idea that "some sample spaces are better than others."

2013-11-30T10:30:52Z

Added reference to additional explanation of multiple sample spaces, specifically the idea that "some sample spaces are better than others."

New page

{{refimprove|date=February 2012}}

{{Probability fundamentals}}
In [[probability theory]], an '''event''' is a [[Set (mathematics)|set]] of [[Outcome (probability)|outcomes]] of an [[Experiment (probability theory)|experiment]] (a [[subset]] of the [[sample space]]) to which a probability is assigned.<ref>{{cite book | last = Leon-Garcia | first = Alberto | title = Probability, Statistics and Random Processes for Electrical Engineering | location = Upper Saddle River, NJ | publisher = Pearson | year = 2008 | url = http://books.google.com/books/about/Probability_Statistics_and_Random_Proces.html?id=GUJosCkbBywC}}</ref> A single outcome may be an element of many different events,<ref>{{cite book | last = Pfeiffer | first = Paul E. | year = 1978 | title = Concepts of probability theory | page = 18 | url = http://books.google.com/books?id=_mayRBczVRwC&pg=PA18 | publisher = Dover Publications | ISBN = 978-0-486-63677-1}}</ref> and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.<ref>{{cite book | last = Foerster | first = Paul A. | year = 2006 | title = Algebra and Trigonometry: Functions and Applications, Teacher's Edition | edition = Classics | page = 634 | publisher = [[Prentice Hall]] | location = Upper Saddle River, NJ | isbn = 0-13-165711-9 | url = http://www.amazon.com/Algebra-Trigonometry-Functions-Applications-Prentice/dp/0131657100}}</ref>

Typically, when the sample space is finite, any subset of the sample space is an event (''i''.''e''. all elements of the [[power set]] of the sample space are defined as events). However, this approach does not work well in cases where the sample space is [[uncountably infinite]], most notably when the outcome is a [[real numbers|real number]]. So, when defining a [[probability space]] it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see ''[[#Events in probability spaces|Events in probability spaces]]'', below).

==A simple example==
If we assemble a deck of 52 [[playing card]]s 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 [[singleton set]] (an [[elementary event]]), the [[empty set]] (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are [[proper subset]]s of the sample space that contain multiple elements. So, for example, potential events include:
[[Image:Venn A subset B.svg|thumb|150px|A [[Venn diagram]] of an event. ''B'' is the sample space and ''A'' is an event.<br>By the ratio of their areas, the probability of ''A'' is approximately 0.4.]]

* "Red and black at the same time without being a joker" (0 elements),
* "The 5 of Hearts" (1 element),
* "A King" (4 elements),
* "A Face card" (12 elements),
* "A Spade" (13 elements),
* "A Face card or a red suit" (32 elements),
* "A card" (52 elements).
Since all events are sets, they are usually written as sets (e.g. {1, 2, 3}), and represented graphically using [[Venn diagram]]s. Given that each outcome in the sample space Ω is equally likely, the probability of an event ''A'' is the following {{visible anchor|formula}}: <br />
:<math>\mathrm{P}(A) = \frac{|A|}{|\Omega|}\,\ ( \text{alternatively:}\ \Pr(A) = \frac{|A|}{|\Omega|})</math>
This rule can readily be applied to each of the example events above.

==Events in probability spaces==
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 [[probability distributions]], such as the [[normal distribution]], the sample space is the set of real numbers or some subset of the [[real numbers]]. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers [[Pathological (mathematics)|'badly behaved']] sets, such as those that are [[nonmeasurable set|nonmeasurable]]. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as [[joint probability|joint]] and [[conditional probability|conditional probabilities]], to work, it is necessary to use a [[sigma-algebra|σ-algebra]], that is, a family closed under complementation and countable unions of its members. The most natural choice is the [[Borel measure|Borel measurable]] set derived from unions and intersections of intervals. However, the larger class of [[Lebesgue measure|Lebesgue measurable]] sets proves more useful in practice.

In the general [[measure theory|measure-theoretic]] description of [[probability space]]s, an event may be defined as an element of a selected [[sigma-algebra|σ-algebra]] 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 ''events of interest'' are elements of the σ-algebra.

==A note on notation==
Even though events are subsets of some sample space Ω, they are often written as [[propositional formula]]s involving [[random variable]]s. For example, if ''X'' is a real-valued random variable defined on the sample space Ω, the event
:<math>\{\omega\in\Omega \mid u < X(\omega) \leq v\}\,</math>
can be written more conveniently as, simply,
:<math>u < X \leq v\,.</math>
This is especially common in formulas for a [[probability]], such as
:<math>\Pr(u < X \leq v) = F(v)-F(u)\,.</math>
The [[set (mathematics)|set]] ''u'' < ''X'' ≤ ''v'' is an example of an [[inverse image]] under the [[map (mathematics)|mapping]] ''X'' because <math>\omega \in X^{-1}((u, v])</math> if and only if <math>u < X(\omega) \leq v</math>.

==See also==
*[[Complementary event]]
*[[Elementary event]]

==Notes==
{{Reflist}}

==External links==
*{{springer|title=Random event|id=p/r077290}}
*[http://mws.cs.ru.nl/mwiki/prob_1.html#M1 Formal definition] in the [[Mizar system]].

{{DEFAULTSORT:Event (Probability Theory)}}
[[Category:Probability theory]]