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The probability $P\left(A\right)$ of an event A is a measure of the likelihood that the event will occur on any trial. Sometimes partial information determines thatan event C has occurred. Given this information, it may be necessary to reassign the likelihood for each event A . This leads to the notion of conditional probability. For a fixed conditioning event C , this assignment to all events constitutes a new probability measure which has all the properties of the originalprobability measure. In addition, because of the way it is derived from the original, the conditional probability measure has a number of special propertieswhich are important in applications.
The original or prior probability measure utilizes all available information to make probability assignments $P\left(A\right),\phantom{\rule{0.277778em}{0ex}}P\left(B\right)$ , etc., subject to the defining conditions (P1), (P2), and (P3) . The probability $P\left(A\right)$ indicates the likelihood that event A will occur on any trial.
Frequently, new information is received which leads to a reassessment of the likelihood of event A . For example
New, but partial, information determines a conditioning event C , which may call for reassessing the likelihood of event A . For one thing, this means that A occurs iff the event $AC$ occurs. Effectively, this makes C a new basic space. The new unit of probability mass is $P\left(C\right)$ . How should the new probability assignments be made? One possibility is to make the new assignment to A proportional to the probability $P\left(AC\right)$ . These considerations and experience with the classical case suggests the following procedure for reassignment. Althoughsuch a reassignment is not logically necessary, subsequent developments give substantial evidence that this is the appropriate procedure.
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