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## Rules of probability

The subjective definition of probability, together with the condition of coherence, requires that . This is one of the rules which probability has to obey. It is possible, in fact, to demonstrate that coherence yields to the standard rules of probability, generally known as axioms. At this point it is worth clarifying the relationship between the axiomatic approach and the others.
• Combinatorial and frequentistic definitions'' give useful rules for evaluating probability, although they do not, as it is often claimed, define the concept.
• In the axiomatic approach one refrains from defining what the probability is and how to evaluate it: probability is just any real number which satisfies the axioms. It is easy to demonstrate that the probabilities evaluated using the combinatorial and the frequentistic prescriptions do in fact satisfy the axioms.
• The subjective approach to probability, together with the coherence requirement, defines what probability is and provides the rules which its evaluation must obey; these rules turn out to be the same as the axioms.

Since everybody is familiar with the axioms and with the analogy (see Table  and Fig. ) let us remind ourselves of the rules of probability in this form:

Table: Events versus sets.
 Events Sets Symbol event set certain event sample space impossible event empty set implication inclusion (subset) opposite event complementary set ( ) (complementary) logical product (AND'') intersection logical sum (OR'') union incompatible events disjoint sets complete class finite partition

Axiom 1
;
Axiom 2
(a certain event has probability 1);
Axiom 3
, if
From the basic rules the following properties can be derived:
1:
;
2:
;
3:
if then ;
4:
.
We also anticipate here another rule which will be discussed in Section :
5:

Next: Subjective probability and objective'' Up: Probability Previous: Subjective definition of probability   Contents
Giulio D'Agostini 2003-05-15