Conditional probability

Introduction

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.

Conditional probability

The original or prior probability measure utilizes all available information to make probability assignments $P\left(A\right),P\left(B\right)$ , 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

• An applicant for a job as a manager of a service department is being interviewed. His résumé shows adequate experience and other qualifications. He conductshimself with ease and is quite articulate in his interview. He is considered a prospect highly likely to succeed. The interview is followed by an extensive backgroundcheck. His credit rating, because of bad debts, is found to be quite low. With this information, the likelihood that he is a satisfactory candidate changes radically.
• A young woman is seeking to purchase a used car. She finds one that appears to be an excellent buy. It looks “clean,” has reasonable mileage, and is a dependablemodel of a well known make. Before buying, she has a mechanic friend look at it. He finds evidence that the car has been wrecked with possible frame damage that has beenrepaired. The likelihood the car will be satisfactory is thus reduced considerably.
• A physician is conducting a routine physical examination on a patient in her seventies. She is somewhat overweight. He suspects that she may be prone to heartproblems. Then he discovers that she exercises regularly, eats a low fat, high fiber, variagated diet, and comes from a family in which survival well into their ninetiesis common. On the basis of this new information, he reassesses the likelihood of heart problems.

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.