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Given two events A and B from the sigma-field of a probability space, with the unconditional probability of B being greater than zero (i.e., P(B) > 0), the conditional probability of A given B (()) is the probability of A occurring if B has or is assumed to have happened. [5]
This rule allows one to express a joint probability in terms of only conditional probabilities. [4] The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.
Often, for some partition {A j} of the sample space, the event space is given in terms of P(A j) and P(B | A j). It is then useful to compute P(B) using the law of total probability: = (), Or (using the multiplication rule for conditional probability), [24]
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.
In this sense, "the concept of a conditional probability with regard to an isolated hypothesis whose probability equals 0 is inadmissible." (Kolmogorov [6]) The additional input may be (a) a symmetry (invariance group); (b) a sequence of events B n such that B n ↓ B, P ( B n) > 0; (c) a partition containing the given event. Measure-theoretic ...
For example, consider the task with coin flipping, but extended to n flips for large n. In the ideal case, given a partial state (a node in the tree), the conditional probability of failure (the label on the node) can be efficiently and exactly computed. (The example above is like this.)
A conditional probability table can be put into matrix form. As an example with only two variables, the values of P ( x 1 = a k ∣ x 2 = b j ) = T k j , {\displaystyle P(x_{1}=a_{k}\mid x_{2}=b_{j})=T_{kj},} with k and j ranging over K values, create a K × K matrix.
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...