248 



Chance, Causation, and Design. [CHAP. x. 



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. 



text, whether this improbability gave in the chapter on Eandomness. We 



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 (fl-) 10 , 

 viz. 8. This represents odds of only 

 about 4 : 1 against such an occur 

 rence, 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 i and 

 Tan- 1 -^-^, show the same diver 

 gencies (v. Proc. Roy. Soc. xxi. 319). 

 I may call attention here to a 

 point which should have been noticed 



must be cautious when we decide 

 upon the random character by mere 

 inspection. It is very instructive 

 here to compare the digits in TT with 

 those within the period of a circu 

 lating decimal of very long period. 

 That of l-f-7699, which yields the 

 full period of 7698 figures, was cal 

 culated some years ago by two Cam 

 bridge graduates (Mr Lunn and Mr 

 Suffield), and privately printed. If we 

 confine our examination to a portion 

 of the succession the random cha 

 racter 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 dis 

 appear. That is, instead of a merely 

 ultimate approximation to equality 

 we should have (as far as this is 

 possible) an absolute attainment 

 of it. 



