360 Prof. Donkin on certain Questions relating to 



arrangement, he voulcl choose that particular arrangement which 

 is observed. Then if we put D for the a posteriori probabihty 

 (resulting from the observed phreuomenon) that the arrangement 

 was designed, we have, by Theorem 11., 



D = 



drp 



drp-\-{l—d)a' 



(It is as well to observe that the two ways in which tlie phseno- 

 menon might be produced have for their a priori probabilities 

 drp, (1 —d)a ; and the three ways in which it might fail, dr{l —p), 

 d[l — r), (1— f/)(l— c/) ; and the sum of these five expressions 

 is 1, as it ought to be.) 



12. In examining the above expression for D, let us first 

 assign a value to d. If we have no knowledge beforehand as to 

 whether the person who placed the balls would be likely to intend 



some arrangement or not, then d= -, and the value of D may be 



written 



1 



D = 



1 + 



ti 



>.':i>.r.-) ■'. lyiCfMR 



in which we observe! that the ratio — is necessarily ver*Jf sittkU. 



Foi', r being the probability that a person designing some arrange- 

 ment would choose a rpyular one, must be taken to be nearly 1, 



or at all events greater than -. And p, the probability of his 



choosing a given regular arrangement out of all possible regular 

 arrangements, is greater than a in the same ratio that the whole 

 laumber of possible arrangements is greater than the whole num- 

 ber of regular arrangements ; a ratio of which both the terms 

 are infinite, but vrhich must certainly be very great, if not itself 

 infinite. If it were absolutely infinite, we should have D = 1 ; 

 and it is unquestionably great enough to make the value of D 

 sensibly equal to 1. Tluis the mathematical investigation leads, 

 equally with comoion sense, to a moral certaintij that the arrange- 

 ment was designed. 



13. The solution of such problems as that which we have just 

 considered, always involves, in its expression, the values of apriori 

 probabilities; that is, probabilities derived from information 

 which we possess antecedently to the observation of the phseno- 

 mcnon considered. Now since every hypothesis has a determi- 

 nate probability corresponding to a determinate state of infor- 

 mation, and since I must be in a detcmiinate state of information 

 with respect to any proposed hypothesis^ it follows that the values 



