The reader must clearly understand that we are not now discussing the mere question of fact whether a certain assigned arrangement _is_ what we call a chance one. This, as was fully pointed out in the fourth chapter, can be settled by mere inspection, provided the materials are extensive enough. What we are now proposing to do is to carry on the enquiry from the point at which we then had to leave it off, by solving the question, Given a certain arrangement, is it more likely that this was _produced_ by design, or by some of the methods commonly called chance methods? The distinction will be obvious if we revert to the succession of figures which constitute the ratio π. As I have said, this arrangement, regarded as a mere succession of digits, appears to fulfil perfectly the characteristics of a chance arrangement. If we were to omit the first four or five digits, which are familiar to most of us, we might safely defy any one to whom it was shown to say that it was not got at by simply drawing figures from a bag. He might look at it for his whole life without detecting that it was anything but the result of such a chance selection. And rightly so, because regarded as a mere arrangement it _is_ a chance one: it fulfils all the requirements of such an arrangement.[2] The question we are now proceeding to discuss is this: Given any such arrangement how are we to determine the process by which it was arrived at? We are supposed to have some event before us which might have been produced in either of two alternative ways, i.e. by chance or by some kind of deliberate design; and we are asked to determine the odds in favour of one or other of these alternatives. It is therefore a problem in Inverse Probability and is liable to all the difficulties to which problems of this class are apt to be exposed. 12. For the theoretic solution of such a question we require the two following data:-- (1) The relative frequency of the two classes of agencies, viz. that which is to act in a chance way and that which is to act designedly. (2) The probability that each of these agencies, if it were the really operative one, would produce the event in question. The latter of these data can generally be secured without any difficulty. The determination of the various contingencies on the chance hypothesis ought not, if the example were a suitable one, to offer any other than arithmetical difficulties. And as regards the design alternative, it is generally taken for granted that if this had been operative it would certainly have produced the result aimed at. For instance, if ten pence are found on a table, all with head uppermost, and it be asked whether chance or design had been at work here; we feel no difficulty up to a certain point. Had the pence been tossed we should have got ten heads only once in 1024 throws; but had they been placed designedly the result would have been achieved with certainty.