It seems plain that we have no experimental knowledge of series piled on series after this fashion. Our conclusions about them are not based on observation, nor collected from statistics. They are arrived at _a priori_; and when the character of a series is arrived at _a priori_, the probability of a particular event belonging to it can be arrived at independently by the same method. No reference to the series is required. The reason we estimate the chances against any one of the six possible throws of a die as five to one under each and all of the suppositions we have been discussing is that under none of them have we any ground for thinking any one of the six more probable than another;—even though we may have ground for thinking that in a series of throws made with that particular die, some number, to us unknown, will in fact turn up with exceptional frequency. The most characteristic examples, therefore, of problems in probability depend for their solution on a bold use of the “principle of sufficient reason.” We treat alternatives as equally likely when we cannot see any ground for supposing that one is more likely than another. This seems sensible enough; but how far may we carry this process of extracting knowledge from ignorance? An agnostic declines to offer any opinion on the being of God because it is a matter about which he professes to know nothing. But the universe either has a spiritual cause, or it has not. If the agnostic is as ignorant as he supposes, he cannot have any reason for preferring the first alternative to the second, or the second to the first. Must he, therefore, conclude that the chances of Theism are even? The man who knows this knows much. He knows, or may know, that God’s existence is slightly more probable than his own chance of winning a coup at Monte Carlo. He knows, or may know, the exact fraction by which the two probabilities differ. How, then, can he call himself an agnostic?