Elementary event
Encyclopedia
In probability theory
, an elementary event
or atomic event is a singleton of a sample space. An outcome is an element of a sample space. An elementary event is a set containing exactly one outcome, not the outcome itself. However, elementary events are often written as outcomes for simplicity when the difference is unambiguous.
The following are examples of elementary events:
Elementary events may have probabilities that are strictly positive, zero, undefined, or any combination thereof. For instance, any discrete probability distribution is determined by the probabilities
it assigns to what may be called elementary events. In contrast, all elementary events have probability zero under any continuous distribution. Mixed distributions, being neither entirely continuous nor entirely discrete, may contain atoms. Atoms can be thought of as elementary (that is, atomic) events with non-zero probabilities. Under the measure-theoretic definition of a probability space
, the probability of an elementary event need not even be defined, since mathematicians distinguish between the sample space S and the events of interest, defined by the elements of a σ-algebra
on S.
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...
, an elementary 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...
or atomic event is a singleton of a sample space. An outcome is an element of a sample space. An elementary event is a set containing exactly one outcome, not the outcome itself. However, elementary events are often written as outcomes for simplicity when the difference is unambiguous.
The following are examples of elementary events:
- All sets {k}, where k ∈ N if objects are being counted and the sample space is S = {0, 1, 2, 3, ...} (the natural numbers).
- {HH}, {HT}, {TH} and {TT} if a coin is tossed twice. S = {HH, HT, TH, TT}. H stands for heads and T for tails.
- The real numbers, and the elementary events, are all sets {x}, where x ∈ R if X is a normally distributed random variableRandom variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
and S = (−∞, +∞). This example shows that, because they are all zero, the probabilities assigned to atomic events do not determine a continuous probability distributionProbability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
.
Elementary events may have probabilities that are strictly positive, zero, undefined, or any combination thereof. For instance, any discrete probability distribution is determined by the probabilities
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...
it assigns to what may be called elementary events. In contrast, all elementary events have probability zero under any continuous distribution. Mixed distributions, being neither entirely continuous nor entirely discrete, may contain atoms. Atoms can be thought of as elementary (that is, atomic) events with non-zero probabilities. Under the measure-theoretic definition of a probability space
Probability space
In 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...
, the probability of an elementary event need not even be defined, since mathematicians distinguish between the sample space S and the events of interest, defined by the elements of 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...
on S.