1.1 Probability systems  (Page 2/4)

— $\square$

This example illustrates several important properties of the classical probability.

1. $P\left(A\right)=N_A/N$ is a nonnegative quantity.
2. $P\left(\Omega \right)=N/N=1$
3. If $A,B,C$ are mutually exclusive , then the number in the disjoint union is the sum of the numbers in the individual events, so that
   $$P(A\bigvee B\bigvee C)=P\left(A\right)+P\left(B\right)+P\left(C\right)$$

Several other elementary properties of the classical probability may be identified. It turns out that they can be derived from these three. Although the classical model is highly useful, andan extensive theory has been developed, it is not really satisfactory for many applications (the communications problem, for example). We seek a more general model which includesclassical probability as a special case and is thus an extension of it. We adopt the Kolmogorov model (introduced by the Russian mathematician A. N. Kolmogorov) which captures the essential ideas in a remarkably successful way.Of course, no model is ever completely successful. Reality always seems to escape our logical nets.

The Kolmogorov model is grounded in abstract measure theory. A full explication requires a level of mathematical sophistication inappropriate for a treatment such as this.But most of the concepts and many of the results are elementary and easily grasped. And many technical mathematical considerations are not important for applications at the levelof this introductory treatment and may be disregarded. We borrow from measure theory a few key facts which are either very plausible or which can be understood at a practical level. This enables us to utilize a very powerful mathematical system for representing practical problems in a manner that leads to both insight and useful strategies of solution .

Our approach is to begin with the notion of events as sets introduced above, then to introduce probability as a number assigned to events subject to certain conditions which becomedefinitive properties. Gradually we introduce and utilize additional concepts to build progressively a powerful and usefuldiscipline. The fundamental properties needed are just those illustrated in [link] for the classical case.