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 ...

, an experiment or trial (see below) is any procedure that can be

infinitely Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...

repeated and has a

well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A funct ...

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 possible outcomes, known as the

sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...

. An experiment is said to be ''random'' if it has more than one possible outcome, and ''deterministic'' if it has only one. A random experiment that has exactly two (

mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...

) possible outcomes is known as a

Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...

. When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of

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 ...

, all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of

statistical analysis Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...

Experiments and trials

Random experiments are often conducted repeatedly, so that the collective results may be subjected to

. A fixed number of repetitions of the same experiment can be thought of as a composed experiment, in which case the individual repetitions are called trials. For example, if one were to toss the same coin one hundred times and record each result, each toss would be considered a trial within the experiment composed of all hundred tosses.

Mathematical description

A random experiment is described or modeled by a mathematical construct known as 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 ...

. A probability space is constructed and defined with a specific kind of experiment or trial in mind. A mathematical description of an experiment consists of three parts: # A

, Ω (or ''S''), which is the

of all possible outcomes. # A set of

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 ...

\scriptstyle \mathcal

, where each event is a set containing zero or more outcomes. # The assignment of probabilities to the events—that is, a function ''P'' mapping from events to probabilities. An ''outcome'' is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complicated ''events'' are used to characterize groups of outcomes. The collection of all such events is a '' sigma-algebra''

\scriptstyle \mathcal

. Finally, there is a need to specify each event's likelihood of happening; this is done using the ''

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...

'' function, ''P''. Once an experiment is designed and established, ''ω'' from the sample space Ω, all the events in

\scriptstyle \mathcal

that contain the selected outcome ''ω'' (recall that each event is a subset of Ω) are said to “have occurred”. The probability function ''P'' is defined in such a way that, if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would

approach Approach may refer to: Aviation *Visual approach *Instrument approach *Final approach Music * ''Approach'' (album), by Von Hertzen Brothers * ''The Approach'', an album by I:Scintilla Other uses *Approach Beach, a gazetted beach in Ting Kau, Ho ...

agreement with the values ''P'' assigns them. As a simple experiment, we may flip a coin twice. The sample space (where the order of the two flips is relevant) is ' where "H" means "heads" and "T" means "tails". Note that each of ''(H, T), (T, H)'', ... are possible ''outcomes'' of the experiment. We may define an ''event'' which occurs when a "heads" occurs in either of the two flips. This event contains all of the outcomes except ''(T, T)''.

References

External links

* {{Authority control

Experiments and trials

Mathematical description

See also

References

External links