Event (probability theory)

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Event" probability theory – news · newspapers · books · scholar · JSTOR (January 2018) (Learn how and when to remove this message)

Part of a series on statistics
Probability theory
Probability Axioms Determinism System Indeterminism Randomness
Probability space Sample space Event Collectively exhaustive events Elementary event Mutual exclusivity Outcome Singleton Experiment Bernoulli trial Probability distribution Bernoulli distribution Binomial distribution Exponential distribution Normal distribution Pareto distribution Poisson distribution Probability measure Random variable Bernoulli process Continuous or discrete Expected value Variance Markov chain Observed value Random walk Stochastic process
Complementary event Joint probability Marginal probability Conditional probability
Independence Conditional independence Law of total probability Law of large numbers Bayes' theorem Boole's inequality
Venn diagram Tree diagram
v t e

In

. An event that has more than one possible outcome is called a compound event. An event ${\displaystyle S}$ is said to occur if ${\displaystyle S}$ contains the outcome ${\displaystyle x}$ of the experiment (or trial) (that is, if ${\displaystyle x\in S}$).^[4] The probability (with respect to some probability measure) that an event ${\displaystyle S}$ occurs is the probability that ${\displaystyle S}$ contains the outcome ${\displaystyle x}$ of an experiment (that is, it is the probability that ${\displaystyle x\in S}$). An event defines a complementary event, namely the complementary set (the event not occurring), and together these define a Bernoulli trial: did the event occur or not?

Typically, when the

, below).

A simple example

If we assemble a deck of 52

proper subsets

of the sample space that contain multiple elements. So, for example, potential events include:

• "Red and black at the same time without being a joker" (0 elements),
• "The 5 of Hearts" (1 element),
• "A King" (4 elements),
• "A Face card" (12 elements),
• "A Spade" (13 elements),
• "A Face card or a red suit" (32 elements),
• "A card" (52 elements).

Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using Venn diagrams. In the situation where each outcome in the sample space Ω is equally likely, the probability ${\displaystyle P}$ of an event ${\displaystyle A}$ is the following formula: $${\displaystyle \mathrm {P} (A)={\frac {|A|}{|\Omega |}}\,\ \left({\text{alternatively:}}\ \Pr(A)={\frac {|A|}{|\Omega |}}\right)}$$ This rule can readily be applied to each of the example events above.

Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard

sets proves more useful in practice.

In the general

𝜎-algebra

of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the 𝜎-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all events of interest are elements of the 𝜎-algebra.

A note on notation

Even though events are subsets of some sample space ${\displaystyle \Omega ,}$ they are often written as predicates or indicators involving random variables. For example, if ${\displaystyle X}$ is a real-valued random variable defined on the sample space ${\displaystyle \Omega ,}$ the event $${\displaystyle \{\omega \in \Omega \mid u<X(\omega )\leq v\}\,}$$ can be written more conveniently as, simply, $${\displaystyle u<X\leq v\,.}$$ This is especially common in formulas for a probability, such as $${\displaystyle \Pr(u<X\leq v)=F(v)-F(u)\,.}$$ The set ${\displaystyle u<X\leq v}$ is an example of an

${\displaystyle X}$ because ${\displaystyle \omega \in X^{-1}((u,v])}$ if and only if ${\displaystyle u<X(\omega )\leq v.}$

See also

• Atom (measure theory) – A measurable set with positive measure that contains no subset of smaller positive measure
• Complementary event – Opposite of a probability event
• Elementary event – Event that contains only one outcome
• Independent event
   – When the occurrence of one event does not affect the likelihood of anotherPages displaying short descriptions of redirect targets
• Outcome (probability) – Possible result of an experiment or trial
• Pairwise independent events – Set of random variables of which any two are independent

External links

Wikimedia Commons has media related to Event (probability theory).

• "Random event", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Formal definition in the Mizar system.