the Theory of Probabilities. 367 



bitants of the house in which the chess-board was seen, and their 

 habits. Suppose him wholly ignorant of these particulars, then 



we may put ff= ^ ^^^^ '^— h- ^^^^ ^^'il^ ^^^^ assume S = ^. With 



respect to i, a person ignorant of the circumstances of the house- 

 hold must use his belief, derived from general experience, that 

 an interruption will occur within any specified period equal to 

 the average duration of a move at chess ; recollecting to avoid the 

 fallacy of substituting for this his a posteriori belief that an in- 

 terruption did occur in this case. The value of i will then cei*- 

 tainly be small. 



His estimate of p depends upon his knowledge of the game; 

 if all possible situations were equally probable, it would hardly be 

 impracticable to assign its numerical value, s will depend upon 



the character of the observed arrangement : and the ratio - will 



° a 



be very large if the arrangement be symmetrical or in any way 

 remarkable, but will approach to unity if the arrangement be 

 irregular. Then we have 



2pi 

 and we obsei-ve that if the arrangement be an irregular one, but 

 much more likely to have occurred in a game than to have resulted 



from either choice or accident, then — and - are both very small, 



.PP. . 



and the value of G will differ little from unity, unless i be a 



quantity of the same order as these ratios. This agrees entirely 

 with common sense; for our judgement must depend upon a 

 comparison of the situation ^\'ith the credibility of an interruption 

 occurring. If we take the extreme case, and suppose an inter- 

 ruption impossible, then i=OandG = 0; that is, the situation 

 not being checkmate, the probability of its occurrence in a game 

 would have no effect in making us believe that a game had been 

 played. 



On the other hand, if we suppose the situation such as could 

 not occur in a game, then j9 = and G = 0. If, again, the situa- 



s 

 tion be symmetrical, and unlikely to occur in a game, then - is 



large, and G is nearly equal to 0, as it ought to be. 



23. If we su))p()se the situation to be checkmate, then the ex- 

 pression for G becomes 



^P 

 in which i' represents the a priori probability that the players 



