Set of all possible outcomes or results of a statistical trial or experiment
Part of a series on statistics
Probability theory
Probability
Axioms
Determinism
System
Indeterminism
Randomness
Probability space
Sample space
Event
Collectively exhaustive events
Elementary event
Mutual exclusivity
Outcome
Singleton
Experiment
Bernoulli trial
Probability distribution
Bernoulli distribution
Binomial distribution
Exponential distribution
Normal distribution
Pareto distribution
Poisson distribution
Probability measure
Random variable
Bernoulli process
Continuous or discrete
Expected value
Variance
Markov chain
Observed value
Random walk
Stochastic process
Complementary event
Joint probability
Marginal probability
Conditional probability
Independence
Conditional independence
Law of total probability
Law of large numbers
Bayes' theorem
Boole's inequality
Venn diagram
Tree diagram
v
t
e
In probability theory, the sample space (also called sample description space,[1]possibility space,[2] or outcome space[3]) of an experiment or random trial is the set of all possible outcomes or results of that experiment.[4] A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points,[5] are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite.[6]
A subset of the sample space is an event, denoted by . If the outcome of an experiment is included in , then event has occurred.[7]
For example, if the experiment is tossing a single coin, the sample space is the set , where the outcome means that the coin is heads and the outcome means that the coin is tails.[8] The possible events are , , , and . For tossing two coins, the sample space is , where the outcome is if both coins are heads, if the first coin is heads and the second is tails, if the first coin is tails and the second is heads, and if both coins are tails.[9] The event that at least one of the coins is heads is given by .
For tossing a single six-sided die one time, where the result of interest is the number of pips facing up, the sample space is .[10]
A well-defined, non-empty sample space is one of three components in a probabilistic model (a probability space). The other two basic elements are a well-defined set of possible events (an event space), which is typically the power set of if is discrete or a σ-algebra on if it is continuous, and a probability assigned to each event (a probability measure function).[11]
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.[12]
^Stark, Henry; Woods, John W. (2002). Probability and Random Processes with Applications to Signal Processing (3rd ed.). Pearson. p. 7. ISBN 9788177583564.
^Forbes, Catherine; Evans, Merran; Hastings, Nicholas; Peacock, Brian (2011). Statistical Distributions (4th ed.). Wiley. p. 3. ISBN 9780470390634.
^Hogg, Robert; Tannis, Elliot; Zimmerman, Dale (December 24, 2013). Probability and Statistical Inference. Pearson Education, Inc. p. 10. ISBN 978-0321923271. The collection of all possible outcomes... is called the outcome space.
^Albert, Jim (1998-01-21). "Listing All Possible Outcomes (The Sample Space)". Bowling Green State University. Retrieved 2013-06-25.
^Soong, T. T. (2004). Fundamentals of probability and statistics for engineers. Chichester: Wiley. ISBN 0-470-86815-5. OCLC 55135988.
^"UOR_2.1". web.mit.edu. Retrieved 2019-11-21.
^Ross, Sheldon (2010). A First Course in Probability(PDF) (8th ed.). Pearson Prentice Hall. p. 23. ISBN 978-0136033134.
^Dekking, F.M. (Frederik Michel), 1946- (2005). A modern introduction to probability and statistics : understanding why and how. Springer. ISBN 1-85233-896-2. OCLC 783259968.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
^"Sample Space, Events and Probability" (PDF). Mathematics at Illinois.
^Larsen, R. J.; Marx, M. L. (2001). An Introduction to Mathematical Statistics and Its Applications (3rd ed.). Upper Saddle River, NJ: Prentice Hall. p. 22. ISBN 9780139223037.
^LaValle, Steven M. (2006). Planning Algorithms(PDF). Cambridge University Press. p. 442.
^"Sample Spaces, Events, and Their Probabilities". saylordotorg.github.io. Retrieved 2019-11-21.
In probability theory, the samplespace (also called sample description space, possibility space, or outcome space) of an experiment or random trial is...
can define a probability space which models the throwing of a die. A probability space consists of three elements: A samplespace, Ω {\displaystyle \Omega...
probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the samplespace. Any specified...
of a random phenomenon in terms of its samplespace and the probabilities of events (subsets of the samplespace). For instance, if X is used to denote...
mathematical function in which the domain is the set of possible outcomes in a samplespace (e.g. the set { H , T } {\displaystyle \{H,T\}} which are the possible...
results is called the samplespace of the experiment, sometimes denoted as Ω {\displaystyle \Omega } . The power set of the samplespace is formed by considering...
short) can be represented by a real vector with positive components. The samplespace of compositional data is a simplex: S D = { x = [ x 1 , x 2 , … , x D...
outcomes of an experiment or random trial that has a restricted or reduced samplespace. The conditional probability can be found by the quotient of the probability...
threatening each other. In orthogonal sampling, the samplespace is divided into equally probable subspaces. All sample points are then chosen simultaneously...
performance metrics that apply to data retrieved from a collection, corpus or samplespace. Precision (also called positive predictive value) is the fraction of...
event, also called an atomic event or sample point, is an event which contains only a single outcome in the samplespace. Using set theory terminology, an...
{\displaystyle {\left\{A_{i}\right\}}} is a finite or countable partition of the samplespace, then E ( X ) = ∑ i E ( X ∣ A i ) P ( A i ) . {\displaystyle \operatorname...
In a random experiment, the probabilities of all possible events (the samplespace) must total to 1— that is, some outcome must occur on every trial. For...
occurs to the total number of trials, i.e., by means not of a theoretical samplespace but of an actual experiment. More generally, empirical probability estimates...
random variable, is a function whose value at any given sample (or point) in the samplespace (the set of possible values taken by the random variable)...
{\displaystyle (\Omega ,F,P)} is a probability space, with samplespace Ω {\displaystyle \Omega } , event space F {\displaystyle F} and probability measure...
The Lunar Sample Laboratory Facility (LSLF) is a repository and laboratory facility at NASA's Lyndon B. Johnson Space Center in Houston, Texas, opened...
probability of 1 entails including all the sample points); however, this distinction becomes important when the samplespace is an infinite set, because an infinite...