1.3.1 Statistical Regularity In order to be useful, a model must enable us to make predictions about the future be- havior of a system, and in order to be predictable, a phenomenon must exhibit regu- larity in its behavior. Many probability models in engineering are based on the fact that averages obtained in long sequences of repetitions (trials) of random experi- ments consistently yield approximately the same value. This property is called statistical regularity. Suppose that the above urn experiment is repeated n times under identical condi- tions. Let No(n), N₁(n), and N₂(n) be the number of times in which the outcomes are balls 0, 1, and 2, respectively, and let the relative frequency of outcome k be defined by Nk(n) fk(n) = n (1.1) By statistical regularity we mean that f(n) varies less and less about a constant value as n is made large, that is, lim fk(n) = Pk. (1.2) The constant pk is called the probability of the outcome k. Equation (1.2) states that the probability of an outcome is the long-term proportion of times it arises in a long se- quence of trials. We will see throughout the book that Eq. (1.2) provides the key con- nection in going from the measurement of physical quantities to the probability models discussed in this book. Figures 1.3 and 1.4 show the relative frequencies for the three outcomes in the above urn experiment as the number of trials n is increased. It is clear that all the relative Relative frequency 0.9 0.8 0.7 □0 Outcome 1 Outcome ◇ 2 Outcome 0.6 0.5 0.4 0.3 13 0.2 10 0.1 000006 10 20 30 40 50 Number of trials Relative frequency 1 0.9 0.8 0.7 0 Outcome 1 Outcome 2 Outcome 0.6 0.5 04 e 0.3 0.2 22 0.1 11 обоюв 10 20 30 40 50 fk(n) Nk(n) n