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In statistics, the conditional probability table (CPT) is defined for a set of discrete and mutually dependent random variables to display conditional probabilities of a single variable with respect to the others (i.e., the probability of each possible value of one variable if we know the values taken on by the other variables).
Given , the Radon-Nikodym theorem implies that there is [3] a -measurable random variable ():, called the conditional probability, such that () = for every , and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called regular if () is a probability measure on (,) for all a.e.
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The table below shows the output from a template call (we'll call the template {{Conditional tables/example 1}}) with different values for {{{variable_foo}}}: Template call Result
In this situation, the event A can be analyzed by a conditional probability with respect to B. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(A|B) [2] or occasionally P B (A).
Conditional probability may be treated as a special case of conditional expectation. Namely, P ( A | X) = E ( Y | X) if Y is the indicator of A. Therefore the conditional probability also depends on the partition α X generated by X rather than on X itself; P ( A | g(X) ) = P (A | X) = P (A | α), α = α X = α g(X).
The conditional probability at any interior node is the average of the conditional probabilities of its children. The latter property is important because it implies that any interior node whose conditional probability is less than 1 has at least one child whose conditional probability is less than 1.
Part of a series on statistics: Probability theory; Probability. Axioms; Determinism. System; Indeterminism; Randomness; Probability space; Sample space; Event ...