Books have been written on the "definition" of probability.
We shall merely note two properties: (*a*) statistical
independence (events must be completely unrelated), and
(*b*) the law of large numbers. This says that if
*p*_{1} is the
probability of getting an event in Class 1 and we observe
that *N*_{1} out of *N* events are in Class 1, then we
have

A common example of direct probability in physics is that in
which one has exact knowledge of a final-state wave function
(or probability density). One such case is that in which we
know in advance the angular distribution *f* (*x*), where
*x* = cos
of a certain scattering experiment, In this example one can
predict with certainty that the number of particles that
leave at an angle *x*_{1} in an interval
*x*_{1} is
*Nf*
(*x*_{1})*x*_{1},
where *N*, the total number of scattered particles, is a very large
number. Note that the function *f(x)* is normalized to unity:

As physicists, we call such a function a distribution function.
Mathematicians call it a probability density function. Note
that an element of probability, *dp*, is