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 ...
, an elementary event, also called an atomic event or sample point, 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 eve ...
which contains only a single
outcome
Outcome may refer to:
* Outcome (probability), the result of an experiment in probability theory
* Outcome (game theory), the result of players' decisions in game theory
* ''The Outcome'', a 2005 Spanish film
* An outcome measure (or endpoint) ...
in the
sample space. Using
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
terminology, an elementary event is a
singleton. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event corresponding to precisely one outcome.
The following are examples of elementary events:
* All sets
where
if objects are being counted and the sample space is
(the
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
).
*
if a coin is tossed twice.
where
stands for heads and
for tails.
* All sets
where
is a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. Here
is a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
and
This example shows that, because the probability of each elementary event is zero, the probabilities assigned to elementary events do not determine a continuous
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
.
Probability of an elementary event
Elementary events may occur with probabilities that are between zero and one (inclusively). In a
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
probability distribution whose sample space is finite, each elementary event is assigned a particular probability. In contrast, in a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
distribution, individual elementary events must all have a probability of zero.
Some "mixed" distributions contain both stretches of continuous elementary events and some discrete elementary events; the discrete elementary events in such distributions can be called atoms or atomic events and can have non-zero probabilities.
Under the
measure-theoretic definition of a
probability space, the probability of an elementary event need not even be defined. In particular, the set of events on which probability is defined may be some
σ-algebra on
and not necessarily the full
power set.
See also
*
*
References
Further reading
*
*
Experiment (probability theory)
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