In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the sample space (also called sample description space, possibility space, or outcome space) of an
experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...
or random
trial
In law, a trial is a coming together of Party (law), parties to a :wikt:dispute, dispute, to present information (in the form of evidence (law), evidence) in a tribunal, a formal setting with the authority to Adjudication, adjudicate claims or d ...
is the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all possible
outcomes or results of that experiment.
A sample space is usually denoted using
set notation
In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
Defining ...
, and the possible ordered outcomes, or sample points,
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
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory inc ...
"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
,
countably
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
infinite, or
uncountably infinite
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
.
A subset of the sample space is an
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of e ...
, denoted by
. If the outcome of an experiment is included in
, then event
has occurred.
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. 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.
The event that at least one of the coins is heads is given by
.
For tossing a single six-sided
die
Die, as a verb, refers to death, the cessation of life.
Die may also refer to:
Games
* Die, singular of dice, small throwable objects used for producing random numbers
Manufacturing
* Die (integrated circuit), a rectangular piece of a semicondu ...
one time, where the result of interest is the number of
pips facing up, the sample space is
.
A well-defined, non-empty sample space
is one of three components in a probabilistic model (a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
). The other two basic elements are: a well-defined set of possible
events
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of ev ...
(an event space), which is typically the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of
if
is discrete or a
σ-algebra on
if it is continuous, and a
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
assigned to each event (a
probability measure function).
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.
Conditions of a sample space
A set
with outcomes
(i.e.
) must meet some conditions in order to be a sample space:
* The outcomes must be mutually exclusive, i.e. if
occurs, then no other
will take place,
.
* The outcomes must be collectively exhaustive, i.e. on every experiment (or random trial) there will always take place some outcome
for
.
* The sample space (
) 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
Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods.
"An abstr ...
must be chosen.
For instance, in the trial of tossing a coin, one possible sample space is
, where
is the outcome where the coin lands heads and
is for tails. Another possible sample space could be
. Here,
denotes a rainy day and
is a day where it is not raining. For most experiments,
would be a better choice than
, as an experimenter likely do not care about how the weather affects the coin toss.
Multiple sample spaces
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
A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a fi ...
s, 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).
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
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
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.
Equally likely outcomes
Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. For any sample space with
equally likely outcomes, each outcome is assigned the probability
. 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
A drawing pin (in British English) or thumb tack (in North American English) is a short nail or pin used to fasten items to a wall or board for display and intended to be inserted by hand, usually using the thumb. A variety of names is used ...
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.
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:
:
For example, if two fair six-sided dice are thrown to generate two
uniformly distributed integers,
and
, each in the range from 1 to 6, inclusive, the 36 possible ordered pairs of outcomes
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
is five is
, 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
, so the probability is
. For a sum of seven, the outcomes in the event are
, so the probability is
.
Simple random sample
In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, inferences are made about characteristics of a
population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
by studying a
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of s ...
of that population's individuals. In order to arrive at a sample that presents an
unbiased estimate
In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In st ...
of the true characteristics of the population, statisticians often seek to study a
simple random sample
In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample ...
—that is, a sample in which every individual in the population is equally likely to be included.
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).
Infinitely large sample spaces
In an elementary approach to
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, any subset of the sample space is usually called an
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of e ...
.
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
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
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 .
See also
*
Parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for th ...
*
Probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
*
Space (mathematics)
*
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or ...
*
Event (probability theory)
In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usua ...
*
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
References
External links
*
{{DEFAULTSORT:Sample Space
Experiment (probability theory)