Axiomatic probability theory, which is the subject of this book, was developed by A. N. Kolmogorov ^[d] in 1933. This theory specifies a set of axioms for a well-defined mathematical model of physical experiments whose outcomes exhibit random variability each time they are performed. The advantage of using a model rather than performing an experiment itself is that it is usually much more efficient in terms of time and money to analyze a mathematical model. This is a sensible approach only if the model correctly predicts the behavior of actual experiments. This is indeed the case for Kolmogorov s theory.

A simple prediction of Kolmogorov s theory arises in the mathematical model for the relative frequency of heads in n tosses of a fair coin that we considered in Example 1.1. In the model of this experiment, the relative frequency converges to 1/2 as n tends to infinity; this is a special case of the the strong law of large numbers, which is derived in Chapter 14. (A related result, known as the weak law of large numbers, is derived in Chapter 3.)

Another prediction of Kolmogorov s theory arises in modeling the situation in Example 1.2. The theory explains why the histogram in Figure 1.4 agrees with the bell-shaped curve overlaying it. In the model, the strong law tells us that for each k, the relative frequency of having exactly k heads in 100 tosses should be close...