Page:Mind (Old Series) Volume 9.djvu/243

 THE PHILOSOPHY OF CHANCE. 231 (whatever their letters) in the above example, 7. The chance of an urn being full or empty is even. Therefore we see by reasoning familiar to students of the Law of Errors, that the chance of 9 or is very small ; and if the number of the urns were infinite, would be infinitesimal ; which is absurd. The answer of course is, that you have no right to frame an hypothesis which is at variance with the experienced fact that one digit does turn up as often as another. " It is making a very unexpected tour," as Hume says of the vagaries of hypothesis, from a supposition erected upon facts to reason down to the negation of the facts. But we do not want to leave the facts at all. We take our stand upon the fact that probability-constants occurring in nature present every variety of fractional value ; and that natural constants in general are found to show no preference for one number rather than another. Acting upon which suppo- sition, while in particular cases we shall err, in the long run we shall find our account. The view here propounded, that the so-called intellectual probability is not essentially different from the probability which is founded upon special statistics, must now be tested by a few examples. (1) The most important instance is that afforded by the Art of Measurement, which postulates that the a priori pro- bability of one value being correct is as great as that of another ; that, as a matter of fact, measurables do in general as often have one value as another. Take the simplest case, that of an instrument of which the errors are known to arrange themselves according to a certain law of error. Given a set of observations, we have to calculate from what point (or value) as centre the given observations are most likely to have diverged. The calculation presupposes that the a priori probability of one value is the same as that of another. That essential fact underlies both the common- sense practice of taking an average, and the most refined methods of the Theory of Observations. (2) Let us take next an example put by Mill. It is known that a box or urn contains a hundred balls of two different colours, white and black, ninety-nine of one colour, and one of the other ; but which colour is in the majority is absolutely unknown. The probability of drawing a white ball is i. I believe that the numerical value is justified by the fact that in a great number of experiences with urns one colour would as often be in the majority as another. And this, whatever the origin of our doubt as to the constitution of the urns ; whether we knew that the urn before us was