Theory of Probability I

On the one hand, as was stated in sec. 5, a priori considerations may have to be brought into account in order to obtain a rough value of a probability. On the other hand, we may use the results of experience to check our a priori estimate of a prob ability. Consider, for example, the throwing of a die. The die is approximately a cube; it has six faces, so that, on the unitary system, we should say that the chance of a 6 being thrown is 1/6, or at any rate is about i/6. But it should be observed that what is really of importance to us, if we are frequently risking money on the throwing of a die, is not the probability, as esti mated by the unitary method, of a particular face appearing, but the proportion of cases in which, in the long run, it actually does appear. If we record these, we may find that the 6 appears rather oftener than we had expected. We can then do either of two things. We can base our probability on the actual expe rience, or we can consider in what respect our original estimate was faulty. We must then take account of the fact that the pips of the die are hollowed out, so that the die is lighter on the 6 side and heavier on the opposite side. It would be very difficult to estimate the effect of the hollowing-out of the pips, and we should thus be driven to the statistical method, i.e., to basing our probability on experience.

It will be assumed, throughout this article, that we are dealing with probability according to the statistical method. If, for example, we say that there are five balls in a box, and that they are all equally likely to be drawn, this is to be interpreted as meaning that in N drawings, where N is very large, each ball is drawn approximately i/5•N times. Similarly, if it is said that the probability that a man will survive for a year is p, this is to be taken as meaning that, out of N men who would be put into the same class, the number who will survive for a year is about pN.

8. "The Long Run" and "Series..

The use of the phrase "in the long run," and the description of a number of events as a " series" (e.g., "a Poisson series"), are not to be taken as im plying a definite arrangement, in time or otherwise. We might, of course, wish to observe the succession of observations, e.g., the succession of colours at roulette; but in this case the suc cession is the event we are observing, and the order in which the individual events occur is not relevant. Phrases such as "in the long run" merely imply that the number of cases observed is very large.

9. Randomness.

In the statement of a problem in probability the expression " random" or " at random" is often used. Thus, in the case of balls in a box it may be stated that "a ball is drawn at random from the box." This may be regarded as merely an indication that we are dealing with a question in probability. On the other hand, the idea of randomness is implied in any state ment as to the numerical measure of a probability. When we

say " If a card is drawn from a pack, the probability that it will be a court card is 3/13," we really mean If a card is drawn at random from a pack, the probability that it will be a court card is 3/13." Io. Addition and Multiplication of Probabilities.—The mathematical treatment of probabilities involves the addition and multiplication of their numerical measures. These are performed according to certain rules.

(i.) The rule for addition is as follows. Let E and E' be two mutually exclusive events which might be associated with C, and let the probabilities of their being so associated be p and p' respectively : then the probability that either E or E' will be so associated is p+ p'. This follows easily from the definition.

(ii.) The

rule for multiplication is that, if C and D are two separate events, both of which occur, and if the probability of C being associated with E is p, and if, in the cases in which C is associated with E, the probability of D being associated with G is r, then the probability that C will be associated with E and D with G is pr. For, if N is the total number of cases in which C and D both occur, the number in which C is associated with E is pN, and the number of these in which D is associated with G is rXpN = prN .

For an illustration of the multiplication rule, we can have recourse to Table I. If we take a father at random (which is the same thing as taking a pair, father and son, at random, since father and son go together), the probability that he will be short is 465/1,000, and, if a father is short, the probability that he has a tall son is 215/465. Hence, if we take a pair, father and son, at random, the probability that the pair consists of a short father and a tall son is 215/1,000, which agrees with the table.

If

r has the same value whether C is associated with E or not, or, to put it differently, if the probability of C being asso ciated with E and D with G is the product of the separate prob abilities of C being associated with E and D with G, the asso ciations of C with E and of D with G are said to be independent.

II. Scope.—The earlier studies of probability led to the development of the theory of combinations and permutations; and the more simple problems of chance are largely concerned with applications of this theory. Many such problems are dealt with in text-books, as well as in previous editions of this work, and it is not necessary to give examples here. The part of the subject with which we are more particularly concerned may be taken as beginning with the theory of error, its stages being marked by the successive inclusion of frequency-distributions generally, and of correlation. Before dealing with these aspects, it is necessary to give brief consideration to the theory of probability of causes.