THEORY OF PROBABILITY I. Rough estimates of relative probability are often easily made. I draw a ball from a box containing one red ball and four white ones: if I say that I am more likely to draw a white than the red, the statement is intelligible. But can I be more precise? Can I give a numerical measure to the probability of drawing the red ball? A question of this kind might perhaps be parried by the question: Why should anyone want a numerical measure of the probability? The answer may be, in some cases, that it is a matter of scientific interest; or it may be, in other cases, that questions of this kind are important in games of chance. The study of the theory of probability was, in fact, in the first in stance, the scientific study of gambling; though the primary question was not the numerical measure of a chance but its money value. Suppose, in the case of the five balls, that I am to receive if I draw the red ball, but nothing if I draw a i white; then (subject to certain conditions) my chance worth Do. This suggests a numerical measure of probability. Sup pose there is a prize W in a lottery, and my chance of winning it is worth V; then V is pw, where p is some fraction between o and 1. If the prize W were doubled, the conditions remaining the same, the value V of the chance would be doubled, and so on.
Thus, the conditions of the lot tery remaining the same, V bears a constant ratio to W; and this ratio, which is a fraction be tween o and 1, might be called the chance, or probability, of my winning the prize. The value of my chance is then found by multiplying the prize by this chance or probability. This definition, however, is not entirely satisfactory; for, the value V being based on, at best, a general agreement, its ratio to W cannot be regarded as a definite objective measure. We need some method of measure ment that is based on facts, not on opinions. Two methods have been suggested (secs. 2 and 4).
and only one, of c ways, all of which are equally likely, and if a of these ways are called favourable, then the probability or chance of the event happening favourably is a/c. Thus, in the case of a ball being drawn from a box containing one red and four white balls, the event is the drawing of a ball, and the ways of the event happening are the drawing of a red ball and the draw ing of one of the white balls; if all the balls are equally likely to be drawn, the probability of the red being drawn is 1/5, and the probability of a white being drawn is 4/5. Since the basis of the method is the supposition that the happening of the event can be subdivided under a number of ways, each of which is equally likely, and the chance of the event happening favourably is found by counting the number of the favourable ways, we can call it the unitary method.
3. Defects of the Method.—While the definition is a simple one as regards a good many problems, it has defects, of which the following may be mentioned.
(i.) It applies only to probabilities of subsequent events, not to those of concurrent or antecedent events (sec. 12).
(ii.) The phrase " equally likely" is not defined. If this means that the probabilities of the different ways are all equal, we are working in a circle. If it means that we do not see any reason why one way should occur rather than another, we are basing the definition on ignorance.
(iii.) There are a great many cases in which the possibility of an event happening cannot be split up into a number of ways all of which are equally likely, however this may be defined. If a box contains an equal number of red balls and white balls, and the red are of the same size as the white but are slightly heavier, the statement that a white is more likely to be drawn than a red is intelligible, but it is difficult to bring the case under the definition. A statement as to probability can here only have a statistical meaning, as explained in sec. 4.