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In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one ...
If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.. If (X, ≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions.
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
Then is a 𝜎-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact A {\displaystyle {\mathcal {A}}} is the minimal 𝜎-field containing R {\displaystyle {\mathcal {R}}} since it must be contained in every 𝜎-field containing R . {\displaystyle {\mathcal {R}}.}
An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers.It is the algebra on which the Borel measure is defined. . Given a real random variable defined on a probability space, its probability distribution is by definition also a measure on the Borel a
In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. [1] A single outcome may be an element of many different events, [2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. [3]
Play the USA TODAY Sudoku Game. JUMBLE. Jumbles: DOUBT QUIRK BANTER VACANT. Answer: Teacher needed the little cats to be serious, but they wouldn't stop - "KITTEN" AROUND
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.