two bags, one with M.s in excess and the other with F.s in less excess, and that some persons draw from one and some from the other. But fuller statistics
It will be observed that the total excess of male births is large. This may arise from undue omission of females; but I have carefully confined myself to the two or three last generations, in each pedigree, for greater security.
1 Essay on Probabilities, p. 114.
2 Doubts have been expressed about the truly random character of the digits in this case (v.[ TN: space] 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.[ TN: space] 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.[ TN: space] 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.[ TN: space] 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.[ TN: space] Tan−1 1/5 and Tan−1 1/239, show the same divergencies (v.[ TN: space] Proc.[ TN: space] Roy.[ TN: space] Soc.[** TN: space] 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.[** TN: space] 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.
3 Of course this conventional estimate is nothing different in kind from that which may attach to any order or succession. Ten heads in succession is intrinsically or objectively indistinguishable in character from alternate heads and tails, or seven heads and three tails, &c. Its distinction only consists in its almost universal acceptance as remarkable.
4 Our Inheritance in the Great Pyramid, Ed. III. 1877.