### 1. DIRECT PROBABILITY

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 p1 is the probability of getting an event in Class 1 and we observe that N1 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 x1 in an interval x1 is Nf (x1)x1, 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