148 THE GRAMMAR OF SCIENCE 



probabilities be represented by the letters C^, C2, C3 . . . 

 etc., then by a well-known law for compounding prob- 

 abilities ^ we shall find that the total probability in 

 favour of a white ball occurring on the (/> + ^ + i )th draw- 

 ing, or of a routine following on p routines and q anomies, 



'^~ Pi Ci-FP, C,-1-P3C3-F. . . 



Now all this is pure calculation ; it involves no new 

 principle, nothing the reader may not take on faith, if he 

 is not an adept in mathematical analysis. We shall there- 

 fore suppose the calculation made "" as Laplace made it, 

 and the result will be found to be that given on our. 

 p. 142, namely, the probability that a white ball will be 

 drawn is ^"^' . Or, since q is either zero or vanishingly 

 small as compared with p, we have the overwhelming prob- 

 ability of the routine of perceptions being maintained on 

 the next trial. 



\ 17. — The Permanency of Routine for the Future 



One particular case is worth noting. Suppose we have 

 experienced in sequences of perceptions which have re- 

 peated themselves n times without any anomy. Sup- 

 pose, further, a new sequence to have repeated itself r 

 times also without anomy. Then in all we have had 

 m{n — I ) -1- r — i repetitions, or cases of routine, and no 

 failures ; hence the probability that the new sequence will 

 repeat itself on the (r-f- i)th occasion is obtained by put- 

 ting /=;//(;?— \)-\-r — I and q = o m. the result of ^ 16, 

 or the odds in favour of a routine occurring on the next 

 occasion with the new sequence are in{ji—\)-\-r to i. 

 Therefore if in and n be very great, there will be over- 

 whelming odds in favour of the new sequence following 



1 The reader will find this law discussed- in any elementary work on 

 algebra. See, for example, Todhunter's Algebra, §§ 732 and 746. 



2 See Todhunter's History of the Theory of Probability, Arts. 374, 847-8 ; 

 Boole's Laws of Thought, chap. xx. § 23 ; or T. Galloway, A Treatise ofi 

 Probability, § v., " On the Probability of Future Events deduced from 

 Experience. " 



