CONDITIONAL PROBABILITY

CONDITIONAL PROBABILITY
Conditional probability

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. (December 2007)

Illustration of conditional probability using a Venn diagram. S is the sample space, and are events. Assuming probability is proportional to area, the unconditional probability P(A) ? 0.33. However, the conditional probability , ? 0.85 and .
In probability theory, the "conditional probability of given " is the probability of if is known to occur. It is commonly denoted , and sometimes . (The vertical line should not be mistaken for logical OR.) can be visualised as the probability of event when the sample space is restricted to event . Mathematically, it is defined for as

Formally, is defined as the probability of according to a new probability function on the sample space, such that outcomes not in have probability 0 and that it is consistent with all original probability measures. The above definition follows (see Formal derivation).[1]
Example
3 Statistical independence
4 Common fallacies 4.1 Assuming conditional probability is of similar size to its inverse
4.2 Assuming marginal and conditional probabilities are of similar size
4.3 Over- or under-weighting priors

5 Formal derivation
6 See also
7 References
8 External links


Definition

[edit] Conditioning on an event

Given two events and in the same probability space with , the conditional probability of given is defined as the quotient of the unconditional joint probability of and , and the unconditional probability of :

The above definition is how conditional probabilities are introduced by Kolmogorov. However, other authors such as De Finetti prefer to introduce conditional probability as an axiom of probability. Although mathematically equivalent, this may be preferred philosophically; under major probability interpretations such as the subjective theory, conditional probability is considered a primitive entity. Further, this "multiplication axiom" introduces a symmetry with the summation axiom[2]:

Multiplication axiom:

Summation axiom (A and B mutually exclusive):

[edit] Definition with ?-algebra

If , then the simple definition of is undefined. However, it is possible to define a conditional probability with respect to a ?-algebra of such events (such as those arising from a continuous random variable).

For example, if X and Y are non-degenerate and jointly continuous random variables with density ƒX,Y(x, y) then, if B has positive measure,