Violations of Probability theory

The Laws of Probability

The axiomatic foundation of probability, and the multitude of theorems derived from it, constitutes the formal—and vast—theory of probability. These, more technical, results are not the subject of this entry. The violations considered here concern the basic laws of probability, which are as follows. Suppose T is an event about which there is uncertainty, such as whether a subject will respond to a [Page 1181]treatment. Let A be another event, such as whether the subject will experience an adverse event while responding to treatment. Denote by P(·) the probability of an event. Thus, P(A) is the probability of an adverse event. If one knows that event T has occurred, then the conditional probability of A given T is written as P(A|T), where the vertical line means “given.” All probabilities, conditional or otherwise, for any events E and F, must conform to the following laws:

• 1.
  0 ≤ P(E) ≤ 1; this is the convexity law.
• 2.
  If E and F are mutually exclusive (the occurrence of one precludes the occurrence of the other), then P(E or F) = P(E) + P(F); this is the addition law.
• 3.
  P(E and F) = P(E|F)P(F) = P(F|E)P(E); this is the multiplication law for the conjunction of E and F.

These laws easily extend to more than two events.

Basic Consequences of the Laws

A useful consequence of the convexity law (1) is that the probability of the opposite or complement of an event E, denoted by Ec, is 1 – P(E). In the addition law (2), note that the “or” in the event “E or F” is not exclusive. The statement of the law proceeds from the assumption that E and F cannot occur simultaneously; that is, they are mutually exclusive. This would hardly be reasonable in the illustration given. One cannot typically preclude the occurrence of an adverse event in a treated subject. In such a case, where E and F are not mutually exclusive, one adjusts by subtracting the probability of their conjunction, a result called the addition

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