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 :