to be accepted on the supposed warrant of mathematics. Revert then to the example of Inverse Probability on p. 182. We saw that under certain assumptions, it would follow that when a single white ball had been drawn from a bag known to contain 10 balls which were white or black, the chance could be determined that there was only one white ball in it. Having done this we readily calculate ‘directly’ the chance that this white ball will be drawn next time. Similarly we can reckon the chances of there being two, three, &c.[** TN: space] up to ten white balls in it, and determine on each of these suppositions the chance of a white ball being drawn next time. Adding these together we have the answer to the question:—a white ball has been drawn once from a bag known to contain ten balls, white or black; what is the chance of a second time drawing
So far only arithmetic is required. For the next step we need higher mathematics, and by its aid we solve this problem:—A white ball has been drawn m times from a
bag which contains any number, we know not what, of balls each of which is white or black, find the chance of the next drawing also yielding a white ball. The answer is
Thus far mathematics. Then comes in the physical assumption that the universe may be likened to such a bag as the above, in the sense that the above rule may be applied to solve this question:—an event has been observed to happen m times in a certain way, find the chance that it will happen in that way next time. Laplace, for instance, has pointed out that at the date of the writing of his Essai Philosophique, the odds in favour of the sun's rising again (on the old assumption as to the age of the world) were 1,826,214 to 1. De Morgan says that a man who standing on the bank of a river has seen ten ships pass by with flags should judge it to be 11 to 1 that the next ship will also carry a flag.
§ 8. It is hard to take such a rule as this seriously, for there does not seem to be even that moderate confirmation of it which we shall find to hold good in the case of the application of abstract formulæ to the estimation of the evidence of witnesses. If however its validity is to be discussed there appear to be two very distinct lines of enquiry along which we may be led.
(1) In the first place we may take it for what it professes to be, and for what it is commonly understood to be, viz.[** TN: space] a rule which assigns the measure of expectation we ought to entertain of the recurrence of the event under the circumstances in question. Of course, on the view adopted in this work, we insist on enquiring whether it is really true that on the average events do thus repeat their performance in accordance with this law. Thus tested, no
one surely would attempt to defend such a formula. So far from past occurrence being a ground for belief in future recurrence, there are (as will be more fully pointed out in the Chapter on Fallacies) plenty of cases in which the direct contrary holds good. Then again a rule of this kind is subject to the very serious perplexity to be explained in our next chapter, arising out of the necessary arbitrariness of such inverse reference. That is, when an event has happened but a few times, we have no certain guide; and when it has happened but once,[2] we have no guide whatever, as to the class of cases to which it is to be referred. In the example above, about the flags, why did we stop short at this notion simply, instead of specifying the size, shape, &c.[** TN: space] of the flags?
De Morgan, it must be remembered, only accepts this rule in a qualified sense. He regards it as furnishing a minimum value for the amount of our expectation. He terms it “the rule of probability of a pure induction,”