sample space

A set $\backslash Omega$ with outcomes $s\_1,\; s\_2,\; \backslash ldots,\; s\_n$ (i.e. $\backslash Omega\; =\; \backslash \{s\_1,\; s\_2,\; \backslash ldots,\; s\_n\backslash \}$) must meet some conditions in order to be a sample space:{{cite web |url = https://ocw.mit.edu/resources/res-6-012-introduction-to-probability-spring-2018/part-i-the-fundamentals |title = Sample Spaces |last=Tsitsiklis |author-link=John Tsitsiklis |first=John |date=Spring 2018 |publisher=Massachusetts Institute of Technology |access-date=July 9, 2018}}

• The outcomes must be mutually exclusive, i.e. if $s\_j$ occurs, then no other $s\_i$ will take place, $\backslash forall\; i,j=1,2,\backslash ldots,n\; \backslash quad\; i\backslash neq\; j$.
• The outcomes must be collectively exhaustive, i.e. on every experiment (or random trial) there will always take place some outcome $s\_i\; \backslash in\; \backslash Omega$ for $i\; \backslash in\; \backslash \{1,\; 2,\; \backslash ldots,\; n\backslash \}$.
• The sample space ($\backslash Omega$) 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 $\backslash Omega\_1\; =\; \backslash \{H,T\backslash \}$, where $H$ is the outcome where the coin lands heads and $T$ is for tails. Another possible sample space could be $\backslash Omega\_2\; =\; \backslash \{(H,R),\; (H,NR),\; (T,R),\; (T,NR)\backslash \}$. Here, $R$ denotes a rainy day and $NR$ is a day where it is not raining. For most experiments, $\backslash Omega\_1$ would be a better choice than $\backslash Omega\_2$, as an experimenter likely does not care about how the weather affects the coin toss.

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 cards, one possibility for the sample space could be the various ranks (Ace through King), while another could be the suits (clubs, diamonds, hearts, or spades).{{cite web |url = https://people.richland.edu/james/lecture/m170/ch05-int.html |title = Stats: Introduction to Probability - Sample Spaces |last=Jones |first=James |year = 1996 |publisher = Richland Community College |access-date = 2013-11-30 }} 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 or upside down, if some cards have been flipped when shuffling.

File:Coin tossing.JPG

File:Brass thumbtack.jpg

{{Main|Equally likely outcomes}}

Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely.{{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 }} For any sample space with $N$ equally likely outcomes, each outcome is assigned the probability $\backslash frac\{1\}\{N\}$.{{Cite web|url=https://www3.nd.edu/~dgalvin1/10120/10120_S16/Topic09_7p2_Galvin.pdf|title=Equally Likely outcomes|website=University of Notre Dame}} 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.{{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}}

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:{{cite book |last1 = Yates |first1 = Daniel S. | last2 = Moore | first2 = David S. |last3 = Starnes |first3 = Daren S. |year = 2003 |title = The Practice of Statistics |edition = 2nd |publisher = 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 }}{{rp|346–347}}

: $\backslash mathrm\{P\}(\backslash text\{event\})\; =\; \backslash frac\{\backslash text\{number\; of\; outcomes\; in\; event\}\}\{\backslash text\{number\; of\; outcomes\; in\; sample\; space\}\}$

For example, if two fair six-sided dice are thrown to generate two uniformly distributed integers, $D\_1$ and $D\_2$, each in the range from 1 to 6, inclusive, the 36 possible ordered pairs of outcomes $(D\_1,D\_2)$ 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 $D\_1\; +\; D\_2$ is five is $\backslash frac\{4\}\{36\}$, since four of the thirty-six equally likely pairs of outcomes sum to five.

If the sample space was 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 $\backslash \{(1,1)\backslash \}$, so the probability is $\backslash frac\{1\}\{36\}$. For a sum of seven, the outcomes in the event are $\backslash \{(1,6),\; (6,1),\; (2,5),\; (5,2),\; (3,4),(4,3)\backslash \}$, so the probability is $\backslash frac\{6\}\{36\}$.{{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}}

{{main|Simple random sample}}

In statistics, inferences are made about characteristics of a population by studying a sample of that population's individuals. In order to arrive at a sample that presents an 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.{{rp|274–275}} The result of this is that every possible combination of individuals who could be chosen for the sample has an equal chance to be the sample that is selected (that is, the space of simple random samples of a given size from a given population is composed of equally likely outcomes).{{Cite web|url=https://web.ma.utexas.edu/users/mks/statmistakes/SRS.html|title=Simple Random Samples|website=web.ma.utexas.edu|access-date=2019-11-21}}

In an elementary approach to probability, any subset of the sample space is usually called an event. However, this gives rise to problems when the sample space is continuous, so that a more precise definition of an event is necessary. Under this definition only measurable subsets of the sample space, constituting a σ-algebra over the sample space itself, are considered events.

An example of an infinitely large sample space is measuring the lifetime of a light bulb. The corresponding sample space would be {{Closed-open|0, ∞}}.

• Parameter space
• Probability space
• Space (mathematics)
• Set (mathematics)
• Event (probability theory)
• σ-algebra

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{{DEFAULTSORT:Sample Space}}

Category:Experiment (probability theory)

Category:Space (mathematics)