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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. This rule allows you to express a joint probability in terms of ...
The conditional probability can be found by the quotient of the probability of the joint intersection of events A and B, that is, (), the probability at which A and B occur together, and the probability of B: [2] [6] [7] = ().
Probability is the branch of mathematics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the ...
In probability, for events A 1, ... the probability of the intersection A I only depends on the cardinality of I, meaning that for every k in {1, ...
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. [1] These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. [2]
The probability that X n = 0 occurs for infinitely many n is equivalent to the probability of the intersection of infinitely many [X n = 0] events. The intersection of infinitely many such events is a set of outcomes common to all of them. However, the sum ΣPr(X n = 0) converges to π 2 /6 ≈ 1.645 < ∞, and so the Borel–Cantelli Lemma ...
v. t. e. 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 ...
The conditional probability based on the intersection of events defined as: = (). [2] satisfies the probability measure requirements so long as () is not zero. [ 3 ] Probability measures are distinct from the more general notion of fuzzy measures in which there is no requirement that the fuzzy values sum up to 1 , {\displaystyle 1,} and the ...
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