366 Prof. Donkin on certain Questions relating to 



rity is allowed to rest upon principles, tliere can be little hope of 

 a settlement of any controverted question. I now proceed briefly 

 to examine one or two other problems suggested by Professor 

 Forbes's paper. And first let us see what account the mathe- 

 matical theory can give of the impression which would be made 

 on a person's mind by seeing several pieces placed in a particular 

 situation on a chess-board. 



21. We will state the problem as follows : — A person who 

 understands the game sees a certain number of pieces placed in 

 a particular situation on the board. What are to him the respect- 

 ive probabilities, — (1), that the situation was actually produced 

 by a game ; (2), that it was produced by some one who (ignorant 

 of the game) designed that situation ; or (3), that the situation 

 was accidental (that is, that the person who placed the pieces 

 on the board did not intend to produce any particular arrangement 

 of them) ? Let it be supposed, for simplicity, that only two hy- 

 potheses are admissible ; namely, that a game has been (wholly 

 or partly) played, and the pieces left undisturbed ; or that the 

 pieces were placed by some one who could have no reference to 

 the rules of the game in placing them, say by a child playing 

 with them. Then let g be the a priori probability that a game 

 ivould be played; and (omitting the words a priori) let/* be the 

 probability that in any one game the observed situation would 

 occm\ 



Further, let us suppose that the situation is not one of check- 

 mate, and that nothing but an accidental interruption would 

 have caused the game to stop short. And let i be the probability 

 that an interruption would occur at any specified move. Also let 

 c be the probability that a child would play with the pieces and 

 place them on the board. 



h the probability that, if so, he would design some arrangement. 



s the probability that in that case he would choose the ob- 

 served situation and number of the pieces. 



a the probability that if he did not design any situation he 

 would have chosen the observed number of pieces, and that the 

 situation would result from accident. Then if we put G, D, A 

 for the a posteriori probabilities of the hypotheses (1.), (3.), (3.) 

 above mentioned, we have 



la--^, -U--^, ^-—^ > 



where 



1,s=gpi-\- c(hs + {\ —h)a^ . 



22. It will be sufficient to examine one of these expressions, 

 say the first of them. The values of _{/ and c would depend upon 

 the knowledge possessed by the person supposed, as to the inha- 



