One of the most striking features of the present day is the steadily increasing use of the ideas of probability theory in a wide variety of scientific fields, involving matters as remote and different as the prediction by geneticists of the relative frequency with which various characteristics occur in groups of individuals, the calculation by telephone engineers of the density of telephone traffic, the maintenance by industrial engineers of manufactured products at a certain standard of quality, the transmission (by engineers concerned with the design of communications and automatic control systems) of signals in the presence of noise, and the study by physicists of thermal noise in electric circuits and the Brownian motion of particles immersed in a liquid or gas. What is it that is studied in probability theory that enables it to have such diverse applications? In order to answer this question, we must first define the property that is possessed in common by phenomena such as the number of individuals possessing a certain genetical characteristic, the number of telephone calls made in a given city between given hours of the day, the standard of quality of the items manufactured by a certain process, the number of automobile accidents each day on a given highway, and so on. Each of these phenomena may often be considered a random phenomenon in the sense of the following definition.

A random (or chance) phenomenon is an empirical phenomenon characterized by the property that its observation under a given set of circumstances does not always lead to the same observed outcome (so that there is no deterministic regularity) but rather to different outcomes in such a way that there is statistical regularity . By this is meant that numbers exist between 0 and 1 that represent the relative frequency with which the different possible outcomes may be observed in a series of observations of independent occurrences of the phenomenon.

Closely related to the notion of a random phenomenon are the notions of a random event and of the probability of a random event. A random event is one whose relative frequency of occurrence, in a very long sequence of observations of randomly selected situations in which the event may occur, approaches a stable limit value as the number of observations is increased to infinity; the limit value of the relative frequency is called the probability of the random event .

In order to bring out in more detail what is meant by a random phenomenon, let us consider a typical random event; namely, an automobile accident. It is evident that just where, when, and how a particular accident takes place depends on an enormous number of factors, a slight change in any one of which could greatly alter the character of the accident or even avoid it altogether. For example, in a collision of two cars, if one of the motorists had started out ten seconds earlier or ten seconds later, if he had stopped to buy cigarettes, slowed down to avoid a cat that happened to cross the road, or altered his course for any one of an unlimited number of similar reasons, this particular accident would never have happened; whereas even a slightly different turn of the steering wheel might have prevented the accident altogether or changed its character completely, either for the better or for the worse. For any motorist starting out on a given highway it cannot be predicted that he will or will not be involved in an automobile accident. Nevertheless, if we observe all (or merely some very large number of) the motorists starting out on this highway on a given day, we may determine the proportion that will have automobile accidents. If this proportion remains the same from day to day, then we may adopt the belief that what happens to a motorist driving on this highway is a random phenomenon and that the event of his having an automobile accident is a random event.

Another typical random phenomenon arises when we consider the experiment of drawing a ball from an urn. In particular, let us examine an urn (or a bowl) containing six balls, of which four are white, and two are red. Except for color, the balls are identical in every detail. Let a ball be drawn and its color noted. We might be tempted to ask “what will be the color of a ball drawn from the urn?” However, it is clear that there is no answer to this question. If one actually performs the experiment of drawing a ball from an urn, such as the one described, the color of the ball one draws will sometimes be white and sometimes red. Thus the outcome of the experiment of drawing a ball is unpredictable.

Yet there are things that are predictable about this experiment. In Table 1A the results of 600 independent trials are given (that is, we have taken an urn containing four white balls and two red balls, mixed the balls well, drawn a ball, and noted its color, after which the ball drawn was returned to the urn; these operations were repeated 600 times). It is seen that in each block of 100 trials (as well as in the entire set of 600 trials) the proportion of experiments in which a white ball is drawn is approximately equal to \(\frac{2}{3}\) . Consequently, one may be tempted to assert that the proportion \(\frac{2}{3}\) has some real significance for this experiment and that in a reasonably long series of trials of the experiment \(\frac{2}{3}\) of the balls drawn will be colored white. If one succumbs to this temptation, then one has asserted that the outcome of the experiment (of drawing a ball from an urn containing six balls, of which four are white and two are red) is a random phenomenon.

More generally, if one believes that the experiment of drawing a ball from an urn will, in a long series of trials, yield a white ball in some definite proportion (which one may not know) of the trials of the experiment, then one has asserted (i) that the drawing of a ball from such an urn is a random phenomenon and (ii) that the drawing of a white ball is a random event.

Let us give an illustration of the way in which one may use the knowledge (or belief) that a phenomenon is random. Consider a group of 300 persons who are candidates for admission to a certain school at which there are facilities for only 200 students. In the interest of fairness it is decided to use a random mechanism to choose the students from among the candidates. In one possible random method the 300 candidates are assembled in a room. Each candidate draws a ball from an urn containing six balls, of which four are white; those who draw white balls are admitted as students. Given an individual student, it cannot be foretold whether or not he will be admitted by this method of selection. Yet, if we believe that the outcome of the experiment of drawing a ball possesses the property of statistical regularity, then on the basis of the experiment represented by Table 1A , which indicates that the probability of drawing a white ball is \(\frac{2}{3}\) , we believe that the number of candidates who will draw white balls, and consequently be admitted as students, will be approximately equal to 200 (note that 200 represents the product of (i) the number of trials of the experiment and (ii) the probability of the event that the experiment will yield a white ball). By a more careful analysis, one can show that the probability is quite high that the number of candidates who will draw white balls is between 186 and 214.

One of the aims of this book is to show how by means of probability theory the same mathematical procedure can be used to solve quite different problems. To illustrate this point, we consider a variation of the foregoing problem which is of great practical interest. Many colleges find that only a certain proportion of the students they admit as students actually enroll. Consequently, a college must decide how many students to admit in order to be sure that enough students will enroll. Suppose that a college finds that only two-thirds of the students it admits enroll; one may then say that the probability is \(\frac{2}{3}\) that a student will enroll. If the college desires to ensure that about 200 students will enroll, it should admit 300 students.

Exercises