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space of events. A comparison may help to distinguish these two levels. The calcu-
lus and the theory of differential equations provide rules and methods for solving
and analyzing differential equations, but the choice of the right equation which fits
the best description of a physical phenomenon is a different thing, which pertains to
the ability of modeling correctly phenomena of a certain type.
The axiomatic approach in probability theory was initiated by Kolmogorov and
was developed within the Russian mathematical tradition. It plays a role which is
comparable to the axiomatic approach in geometry, and it is important for under-
standing the logical basis of probability. However, it cannot replace the intuition and
the analysis of adequacy of probabilistic models with respect to the realities they ap-
ply to. Logical analyses of probability are related to inductive logics and to many
non-classical methods of logical inference (Carnap and Hintikka are two logicians
of the last century who developed theories in this direction). From the mathematical
point of view, probability theory is part of a general field of mathematics, referred
to as
Measure theory
, initiated by French mathematicians of the last century. An im-
portant exponent of this school, Emil Borel, proved the following result about real
numbers, which is related to the intrinsic random character of real numbers:
Almost
all numbers, when expressed in any base, contain every possible digit or possible
string of digits
. Here,
almost all
means that the set of numbers without this property
are a set having null measure (a non-empty set may have null measure).
An event can be expressed by means of a proposition asserting that the value of
avariable
X
belongs to a subset of its range, denoted by range
(
X
)
. This means that
the usual propositional operations
are defined on events. The
conditional
probability
of an event
A
, given an event
B
, is denoted by
P
¬,∧,∨
. It expresses a
sort of
implicational
probability, or the probability of
A
, under the assumption that
event
B
has occurred. Formally:
(
A
|
B
)
P
(
A
∧
B
)
P
(
A
|
B
)=
.
P
(
B
)
Events
A
and
B
are said to be
independent
, and we write
A
||
B
,if
P
(
A
|
B
)=
P
(
A
)
.
Events
A
and
B
are
disjoint
if
P
0. The following rules connect proposi-
tional operations to probabilities . Proposition
(
A
∧
B
)=
¬
A
has to be considered in terms of
=(
∈
)
¬
=(
∈
(
)
−
)
complementary set, that is, if
A
X
S
,then
A
X
range
X
S
.
1.
P
(
A
)
≥
0
2.
P
(
A
∨¬
A
)=
1
3.
P
(
A
∧¬
A
)=
0
4.
P
(
¬
A
)=
1
−
P
(
A
)
5.
P
(
A
∨
B
)=
P
(
A
)+
P
(
B
)
−
P
(
A
∧
B
)
6.
P
(
A
|
B
)=
P
(
A
∧
B
)
/
P
(
B
)
7.
A
.
The theory of probability is the field were even professional mathematicians can
be easily wrong, and very often reasoning under probabilistic hypotheses is very
slippery. Let us report some examples, from [215], which can help to grasp main
subtle points about probability. Assume that a pilot has a 2% chance of being shot
||
B
⇔
P
(
A
∧
B
)=
P
(
A
)
P
(
B
)
