  In previous work we defined and studied a notion of typicality, originated
with B. Russell, for properties and objects in the context of general infinite
first-order structures. In this paper we consider this notion in the context of
finite structures. In particular we define the typicality degree of a property
$\phi(x)$ over finite $L$-structures, for a language $L$, as the limit of the
probability of $\phi(x)$ to be typical in an arbitrary $L$-structure ${\cal M}$
of cardinality $n$, when $n$ goes to infinity. This poses the question whether
the 0-1 law holds for typicality degrees for certain kinds of languages. One of
the results of the paper is that, in contrast to the classical well-known fact
that the 0-1 law holds for the sentences of every relational language, the 0-1
law fails for degrees of properties of relational languages containing unary
predicates. On the other hand it is shown that the 0-1 law holds for degrees of
some basic properties of graphs, and this gives rise to the conjecture that the
0-1 law holds for relational languages without unary predicates. Another theme
is the ``neutrality'' degree of a property $\phi(x)$ ( i.e., the fraction of
$L$-structures in which neither $\phi$ nor $\neg \phi$ is typical), and in
particular the ``regular'' properties (i.e., those with limit neutrality degree
$0$). All properties we dealt with, either of a relational or a functional
language, are shown to be regular, but the question whether {\em every} such
property is regular is open.
