Editing Sample space (section)

{{Short description|Set of all possible outcomes or results of a statistical trial or experiment}}
{{Probability fundamentals}}

In [[probability theory]], the '''sample space''' (also called '''sample description space''',<ref>{{cite book |last1 = Stark |first1 = Henry |last2 = Woods |first2 = John W. |year = 2002 |title = Probability and Random Processes with Applications to Signal Processing |edition = 3rd |publisher = Pearson |page = 7 |isbn = 9788177583564 }}</ref> '''possibility space''',<ref>{{cite book |last1 = Forbes |first1 = Catherine |last2 = Evans |first2 = Merran | last3 = Hastings |first3 = Nicholas |last4 = Peacock |first4 = Brian |year = 2011 |title = Statistical Distributions |url = https://archive.org/details/statisticaldistr00cfor |url-access = limited |edition = 4th |publisher= Wiley |page = [https://archive.org/details/statisticaldistr00cfor/page/n17 3] |isbn = 9780470390634 }}</ref> or '''outcome space'''<ref>{{cite book |last1=Hogg |first1=Robert |last2=Tannis |first2=Elliot |last3=Zimmerman |first3=Dale |date=December 24, 2013 |title=Probability and Statistical Inference |publisher=Pearson Education, Inc |page=10 |isbn=978-0321923271 |quote=The collection of all possible outcomes... is called the outcome space.}}</ref>) of an [[experiment (probability theory)|experiment]] or random [[trial and error|trial]] is the [[Set (mathematics)|set]] of all possible [[Outcome (probability)|outcomes]] or results of that experiment.<ref name="albert">{{cite web |url = http://www-math.bgsu.edu/~albert/m115/probability/sample_space.html |title = Listing All Possible Outcomes (The Sample Space) |last=Albert |first=Jim |date =  1998-01-21 |publisher= Bowling Green State University |access-date = 2013-06-25 }}</ref> A sample space is usually denoted using [[set notation]], and the possible ordered outcomes, or sample points,<ref name=":2">{{Cite book|last=Soong|first=T. T.|url=https://www.worldcat.org/oclc/55135988|title=Fundamentals of probability and statistics for engineers|date=2004|publisher=Wiley|isbn=0-470-86815-5|location=Chichester|oclc=55135988}}</ref> are listed as [[Element (mathematics)|elements]] in the set. It is common to refer to a sample space by the labels ''S'', Ω, or ''U'' (for "[[Universe (mathematics)|universal set]]"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be [[Finite set|finite]], [[Countable set|countably]] infinite, or [[Uncountable set|uncountably infinite]].<ref name=":0">{{Cite web|url=https://web.mit.edu/urban_or_book/www/book/chapter2/2.1.html|title=UOR_2.1|website=web.mit.edu|access-date=2019-11-21}}</ref>

A [[subset]] of the sample space is an [[Event (probability theory)|event]], denoted by <math>E</math>. If the outcome of an experiment is included in <math>E</math>, then event <math>E</math> has occurred.<ref>{{Cite book|last=Ross|first=Sheldon|url=http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf|title=A First Course in Probability|year=2010 |publisher=[[Pearson Prentice Hall]]|isbn=978-0136033134|pages=23|edition=8th }}</ref>

For example, if the experiment is tossing a single coin, the sample space is the set <math>\{H,T\}</math>, where the outcome <math>H</math> means that the coin is heads and the outcome <math>T</math> means that the coin is tails.<ref>{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|last=Dekking, F.M. (Frederik Michel), 1946-|date=2005|publisher=Springer|isbn=1-85233-896-2|oclc=783259968}}</ref> The possible events are <math>E=\{\}</math>, <math>E=\{H\}</math>, <math>E = \{T\}</math>, and <math>E = \{H,T\}</math>. For tossing two coins, the sample space is <math>\{HH, HT, TH, TT\}</math>, where the outcome is <math>HH</math> if both coins are heads, <math>HT
</math> if the first coin is heads and the second is tails, <math>TH</math> if the first coin is tails and the second is heads, and <math>TT</math> if both coins are tails.<ref name=":1">{{Cite web|url=https://faculty.math.illinois.edu/~kkirkpat/SampleSpace.pdf|title=Sample Space, Events and Probability|website=Mathematics at Illinois}}</ref> The event that at least one of the coins is heads is given by <math>E = \{HH,HT,TH\}</math>.

For tossing a single six-sided [[Dice|die]] one time, where the result of interest is the number of [[Pip (counting)|pips]] facing up, the sample space is <math>\{1,2,3,4,5,6\}</math>.<ref>{{cite book |last1 = Larsen |first1 = R. J. |last2 = Marx |first2 = M. L. |year = 2001 | title = An Introduction to Mathematical Statistics and Its Applications |edition = 3rd |publisher = [[Prentice Hall]] |location = Upper Saddle River, NJ |page = 22 |isbn = 9780139223037 }}</ref>

A well-defined, non-empty sample space <math>S</math> is one of three components in a probabilistic model (a [[probability space]]). The other two basic elements are a well-defined set of possible [[Event (probability theory)|events]] (an event space), which is typically the [[power set]] of <math>S</math> if <math>S</math> is discrete or a [[Σ-algebra|&sigma;-algebra]] on <math>S</math> if it is continuous, and a [[probability]] assigned to each event (a [[probability measure]] function).<ref>{{Cite book|last=LaValle|first=Steven M.|url=http://lavalle.pl/planning/ch9.pdf|title=Planning Algorithms|publisher=[[Cambridge University Press]]|year=2006|pages=442}}</ref>
[[File:Sample space.png|thumb|263x263px|A visual representation of a finite sample space and events. The red oval is the event that a number is odd, and the blue oval is the event that a number is prime.]]
A sample space can be represented visually by a rectangle, with the outcomes of the sample space denoted by points within the rectangle. The events may be represented by ovals, where the points enclosed within the oval make up the event.<ref>{{Cite web|url=https://saylordotorg.github.io/text_introductory-statistics/s07-01-sample-spaces-events-and-their.html|title=Sample Spaces, Events, and Their Probabilities|website=saylordotorg.github.io|access-date=2019-11-21}}</ref>