Sample space
Encyclopedia
In probability theory
, the sample space or universal sample space, often denoted S, Ω, or U (for "universe
"), of an experiment
or random trial
is the set of all possible outcomes. For example, if the experiment is tossing a coin, the sample space is the set {head, tail}. For tossing two coins, the sample space is {(head,head), (head,tail), (tail,head), (tail,tail)}. For tossing a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
For some kinds of experiments, there may be two or more plausible sample spaces available. For example, when drawing a card from a standard deck of 52 playing card
s, one possibility for the sample space could be the rank (Ace through King), while another could be the suit
(clubs, diamonds, hearts, or spades). A complete description of outcomes, however, would 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.
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 infinite, so that a more precise definition of 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.
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.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the sample space or universal sample space, often denoted S, Ω, or U (for "universe
Universe (mathematics)
In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation...
"), of an experiment
Experiment (probability theory)
An experiment is any procedure that can be infinitely repeated and has a well-defined set of outcomes. Examples include tossing a coin or rolling a die.In probability theory, an experiment refers to random experiment. An experiment is said to be random experiment if it has more than one possible...
or random trial
Trial and error
Trial and error, or trial by error, is a general method of problem solving, fixing things, or for obtaining knowledge."Learning doesn't happen from failure itself but rather from analyzing the failure, making a change, and then trying again."...
is the set of all possible outcomes. For example, if the experiment is tossing a coin, the sample space is the set {head, tail}. For tossing two coins, the sample space is {(head,head), (head,tail), (tail,head), (tail,tail)}. For tossing a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
For some kinds of experiments, there may be two or more plausible sample spaces available. For example, when drawing a card from a standard deck of 52 playing card
Playing card
A playing card is a piece of specially prepared heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic, marked with distinguishing motifs and used as one of a set for playing card games...
s, one possibility for the sample space could be the rank (Ace through King), while another could be the suit
Suit (cards)
In playing cards, a suit is one of several categories into which the cards of a deck are divided. Most often, each card bears one of several symbols showing to which suit it belongs; the suit may alternatively or in addition be indicated by the color printed on the card...
(clubs, diamonds, hearts, or spades). A complete description of outcomes, however, would specify both the denomination and the suit, and a sample space describing each individual card can be constructed as the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the two sample spaces noted above.
In an elementary approach to probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, any subset of the sample space is usually called an event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...
. However, this gives rise to problems when the sample space is infinite, so that a more precise definition of event is necessary. Under this definition only measurable
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
subsets of the sample space, constituting a σ-algebra
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
over the sample space itself, are considered events.
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.
See also
- Probability spaceProbability spaceIn probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
- Space (mathematics)
- Set (mathematics)
- Event (probability theory)Event (probability theory)In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...