{{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> or '''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 (8th Edition)|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 wasis heads and the outcome <math>T</math> means that the coin wasis 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> and, <math>E = \{T\}</math>., and <refmath>E name=":1" \{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|σ-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>
* The sample space (<math>\Omega</math>) must have the '''right granularity''' depending on what the experimenter is interested in. Irrelevant information must be removed from the sample space and the right [[abstraction]] must be chosen.
For instance, in the trial of tossing a coin, one possible sample space is <math>\Omega_1 = \{H,T\}</math>, where <math>H</math> is the outcome where the coin lands heads and <math>T</math> is for tails. Another possible sample space could be <math>\Omega_2 = \{(H,R), (H,NR), (T,R), (T,NR)\}</math>. Here, <math>R</math> denotes a rainy day and <math>NR</math> is a day where it is not raining. For most experiments, <math>\Omega_1</math> would be a better choice than <math>\Omega_2</math>, as an experimenter likely dodoes not care about how the weather affects the coin toss.
== Multiple sample spaces ==
Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely.<ref>{{cite book |last = Foerster |first = Paul A. |year = 2006 |title = Algebra and Trigonometry: Functions and Applications, Teacher's Edition |edition = Classics |page = [https://archive.org/details/algebratrigonome00paul_0/page/633 633] |publisher = [[Prentice Hall]] |isbn = 0-13-165711-9 |url = https://archive.org/details/algebratrigonome00paul_0/page/633 }}</ref> For any sample space with <math>N</math> equally likely outcomes, each outcome is assigned the probability <math>\frac{1}{N}</math>.<ref>{{Cite web|url=https://www3.nd.edu/~dgalvin1/10120/10120_S16/Topic09_7p2_Galvin.pdf|title=Equally Likely outcomes|website=University of Notre Dame}}</ref> However, there are experiments that are not easily described by a sample space of equally likely outcomes—for example, if one were to toss a [[thumb tack]] many times and observe whether it landed with its point upward or downward, there is no physical symmetry to suggest that the two outcomes should be equally likely.<ref>{{Cite web|url=https://www.coconino.edu/resources/files/pdfs/academics/arts-and-sciences/MAT142/Chapter_3_Probability.pdf|title=Chapter 3: Probability|website=Coconino Community College}}</ref>
Though most random phenomena do not have equally likely outcomes, it can be helpful to define a sample space in such a way that outcomes are at least approximately equally likely, since this condition significantly simplifies the computation of probabilities for events within the sample space. If each individual outcome occurs with the same probability, then the probability of any event becomes simply:<ref name="yates">{{cite book |lastlast1 = Yates |firstfirst1 = Daniel S. | last2 = Moore | first2 = David S. |last3 = Starnes |first3 = Daren S. |year = 2003 |title = The Practice of Statistics |edition = 2nd |publisher = [[W. H. Freeman and Company|Freeman]] |location = New York | url = http://bcs.whfreeman.com/yates2e/ | isbn = 978-0-7167-4773-4 |url-status = dead |archive-url = https://web.archive.org/web/20050209001108/http://bcs.whfreeman.com/yates2e/ |archive-date = 2005-02-09 }}</ref>{{rp|346–347}}
: <math>\mathrm{P}(\text{event}) = \frac{\text{number of outcomes in event}}{\text{number of outcomes in sample space}}</math>
For example, if two fair six-sided dice are thrown to generate two [[Discrete uniform distribution|uniformly distributed]] integers, <math>D_1</math> and <math>D_2</math>, each in the range from 1 to 6, inclusive, the 36 possible ordered pairs of outcomes <math>(D_1,D_2)</math> constitute a sample space of equally likely events. In this case, the above formula applies, such as calculating the probability of a particular sum of the two rolls in an outcome. The probability of the event that the sum <math>D_1 + D_2</math> is five is <math>\frac{4}{36}</math>, since four of the thirty-six equally likely pairs of outcomes sum to five.
If the sample space was the all of the possible sums obtained from rolling two six-sided dice, the above formula can still be applied because the dice rolls are fair, but the number of outcomes in a given event will vary. A sum of two can occur with the outcome <math>\{(1,1)\}</math>, so the probability is <math>\frac{1}{36}</math>. For a sum of seven, the outcomes in the event are <math>\{(1,6), (6,1), (2,45), (45,2), (3,4),(4,3)\}</math>, so the probability is <math>\frac{6}{36}</math>.<ref>{{Cite web|title=Probability: Rolling Two Dice|url=http://www.math.hawaii.edu/~ramsey/Probability/TwoDice.html|access-date=2021-12-17|website=www.math.hawaii.edu}}</ref>
=== Simple random sample ===
== External links ==
* {{Commons category-inline}}
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{{DEFAULTSORT:Sample Space}}
[[Category:Experiment (probability theory)]]
[[Category:Space (mathematics)]]
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