is a set, called
the
probability, or the sample space. We
interpret as a mathematical model listing all relevant outcomes of an
experiment.
Let 
be a 
-field of its subsets, called the
events . Events 
model sentences about the
outcomes of the experiment to which we want to assign probabilities. Under this
interpretation, the union 
of events corresponds to the
alternative, the intersection 
corresponds to the conjunction of
sentences, and the complement A' corresponds to the negation of a
sentence. For , 
denotes the
set-theoretical difference.
For an event 
the probability 
is a number interpreted as
the degree of certainty (in unique experiments), or asymptotic frequency of A
(in repeated experiments). Probability is assigned to all events
, but it must satisfy certain requirements (axioms). A set function
is a probability measure on 
, if it fulfills the following conditions:
Probability axioms do not determine the probabilities in a unique way. The axioms provide only minimal consistency requirements, which are satisfied by many different models.
.
are such that 
and 
are
