2. Doubts have been expressed about the truly random character of the digits in this case (v. De Morgan, _Budget of Paradoxes_, p. 291), and Jevons has gone so far as to ask (_Principles of Science_, p. 529), "Why should the value of π, when expressed to a great number of figures, contain the digit 7 much less frequently than any other digit!" I do not quite understand what this means. If such a question were asked in relation to any unusual divergence from the _à priori_ chance in a case of throwing dice, say, we should probably substitute for it the following, as being more appropriate to our science:--Assign the degree of improbability of the event in question; i.e. its statistical rarity. And we should then proceed to judge, in the way indicated in the text, whether this improbability gave rise to any grounds of suspicion. The calculation is simple. The actual number of 7's, in the 708 digits, is 53: whilst the fair average would be 71. The question is, What is the chance of such a departure from the average in 708 turns? By the usual methods of calculation (v. Galloway on _Probability_) the chances against an excess or defect of 18 are about 44 : 1, in respect of any specified digit. But of course what we want to decide are the chances against _some one of the ten_ showing this divergence. This I estimate as being approximately determined by the fraction (44/45)^{10}, viz. 0.8. This represents odds of only about 4 : 1 against such an occurrence, which is nothing remarkable. As a matter of fact several digits in the two other magnitudes which Mr Shanks had calculated to the same length, viz. Tan^{-1} 1/5 and Tan^{-1} 1/239, show the same divergencies (v. _Proc. Roy. Soc._ xxi. 319). I may call attention here to a point which should have been noticed in the chapter on Randomness. We must be cautious when we decide upon the random character by mere inspection. It is very instructive here to compare the digits in π with those within the 'period' of a circulating decimal of very long period. That of 1 ÷ 7699, which yields the full period of 7698 figures, was calculated some years ago by two Cambridge graduates (Mr Lunn and Mr Suffield), and privately printed. If we confine our examination to a portion of the succession the random character seems plausible; i.e. the digits, and their various combinations, come out in nearly, but not exactly, equal numbers. So if we take batches of 10; the averages hover nicely about 45. But if we took the whole period which 'circulates,' we should find these characteristics overdone, and the random character would disappear. That is, instead of a merely ultimate approximation to equality we should have (as far as this is possible) an absolute attainment of it.