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| | My name's Tommy Youngblood but everybody calls me Tommy. I'm from Denmark. I'm studying at the university (final year) and I play the Viola for 3 years. Usually I choose songs from the famous films ;). <br>I have two sister. I love Hiking, watching movies and Fishing.<br><br>My website: [https://www.facebook.com/LindaAllenYeastInfectionTreatment Linda Allen Yeast Infection Treatment] |
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| {{Probability fundamentals}}
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| In [[probability theory]], the '''sample space''' 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=21 January 1998 |publisher= Bowling Green State University |accessdate=June 25, 2013}}</ref> A sample space is usually denoted using [[set notation]], and the possible outcomes 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]]").
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| For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}. For tossing a single six-sided [[Dice|die]], the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).<ref>{{cite book | last = Larsen | first = R. J. | last2 = Marx | first2 = M. L. | year = 2001 | title = An Introduction to Mathematical Statistics and Its Applications | edition = Third | publisher = [[Prentice Hall]] | location = Upper Saddle River, NJ | page = 22 | url = http://www.pearsonhighered.com/educator/academic/product/0,,0139223037,00%2Ben-USS_01DBC.html | isbn = 9780139223037}}</ref>
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| A well-defined sample space is one of three basic elements in a probabilistic model (a [[probability space]]); the other two are a well-defined set of possible [[Event (probability theory)|events]] (a [[sigma-algebra]]) and a [[probability]] assigned to each event (a [[probability measure]] function).
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| ==Multiple sample spaces==
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| For many experiments, there may be more than one plausible sample space available, depending on what result is of interest to the experimenter. For example, when drawing a card from a standard deck of fifty-two [[playing card]]s, one possibility for the sample space could be the various ranks (Ace through King), while another could be the [[Suit (cards)|suits]] (clubs, diamonds, hearts, or spades).<ref name="albert" /><ref name="jones">{{cite web |url=https://people.richland.edu/james/lecture/m170/ch05-int.html |title=Stats: Introduction to Probability - Sample Spaces |last=Jones |first=James |date=1996 |publisher= Richland Community College |accessdate=November 30, 2013}}</ref> A more complete description of outcomes, however, could specify both the denomination and the suit, and a sample space describing each individual card can be constructed as the [[Cartesian product]] of the two sample spaces noted above (this space would contain fifty-two equally likely outcomes). Still other sample spaces are possible, such as {right-side up, up-side down} if some cards have been flipped in shuffling.
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| ==Equally likely outcomes==
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| [[File:Brass thumbtack.jpg | thumb | 120px | alt = A brass tack with point downward | Up or down? Flipping a brass tack leads to a '''sample space''' composed of two outcomes that are not equally likely.]]
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| In some sample spaces, it is reasonable to estimate or assume that all outcomes in the space are equally likely (that they occur with equal [[probability]]). For example, when tossing an ordinary coin, one typically assumes that the outcomes "head" and "tail" are equally likely to occur. An implicit assumption that all outcomes in the sample space are equally likely underpins most [[randomization]] tools used in common [[games of chance]] (e.g. rolling [[dice]], shuffling [[Playing card|cards]], spinning tops or wheels, drawing [[Lottery|lots]], etc.). Of course, players in such games can try to cheat by subtly introducing systematic deviations from equal likelihood (e.g. with [[Card marking|marked cards]], [[Dice#Loaded_dice|loaded]] or shaved dice, and other methods).
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| 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 = 633 | 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> 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 symmetry to suggest that the two outcomes should be equally likely.
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| 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 | last = Yates | first = 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}}</ref>{{rp|346-347}}
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| :<math>P(event) = \frac{\text{number of outcomes in event}}{\text{number of outcomes in sample space}}</math>
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| ===Simple random sample===
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| {{main|Simple random sample}}
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| In [[statistics]], inferences are made about characteristics of a [[Statistical population|population]] by studying a [[Sample (statistics)|sample]] of that population's individuals. In order to arrive at a sample that presents an [[Bias of an estimator|unbiased estimate]] of the true characteristics of the population, statisticians often seek to study a [[simple random sample]]— that is, a sample in which every individual in the population is equally likely to be included.<ref name="yates" />{{rp|274-275}} The result of this is that every possible combination of individuals who could be chosen for the sample is also equally likely (that is, the space of simple random samples of a given size from a given population is composed of equally likely outcomes).
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| ==Infinitely large sample spaces==
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| In an elementary approach to [[probability]], any subset of the sample space is usually called an [[event (probability theory)|event]]. However, this gives rise to problems when the sample space is [[Infinite set|infinite]], so that a more precise definition of an event is necessary. Under this definition only [[measure (mathematics)|measurable]] subsets of the sample space, constituting a [[sigma-algebra|σ-algebra]] over the sample space itself, are considered events.
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| However, this has essentially only theoretical significance, since in general the σ-algebra can always be defined to include all subsets of interest in applications.
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| ==See also==
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| *[[Probability space]]
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| *[[Space (mathematics)]]
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| *[[Set (mathematics)]]
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| *[[Event (probability theory)]]
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| *[[σ-algebra]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Sample Space}}
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| [[Category:Probability theory]]
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My name's Tommy Youngblood but everybody calls me Tommy. I'm from Denmark. I'm studying at the university (final year) and I play the Viola for 3 years. Usually I choose songs from the famous films ;).
I have two sister. I love Hiking, watching movies and Fishing.
My website: Linda Allen Yeast Infection Treatment