98 Randomness and its scientific treatment. [CHAP. v. 



and the question is to find the ultimate proportion of those 

 which meet a boundary line to the total of those which fall. 

 The problem therefore becomes a merely geometrical one, 

 viz. to determine the ratio of a certain area on the board to 

 the whole area. The determination of this ratio is all that 

 the mathematician ever takes into account. 



Now take the following. A straight brittle rod is broken 

 at random in two places: find the chance that the pieces can 

 make a triangle 1 . Since the only condition for making a 

 triangle with three straight lines is that each two shall be 

 greater than the third, the problem seems to involve the 

 same general conception as in the former case. We must 

 conceive such rods breaking at one pair of spots after 

 another, no one can tell precisely where, but showing the 

 same ultimate tendency to distribute these spots throughout 

 the whole length uniformly. As in the last case, the mathe 

 matician thinks of nothing but this final result, and pays no 

 heed to the process by which it may be brought about. Ac 

 cordingly the problem is again reduced to one of mensura 

 tion, though of a somewhat more complicated character. 



3. (2) In another class of cases we have to contemplate 

 an intermediate process rather than a final result; but the 

 same conception has to be introduced here, though it is now 

 applied to the former stage, and in consequence will not in 

 general apply to the latter. 



For instance : a shot is fired at random from a gun whose 

 maximum range (i.e. at 45 elevation) is 3000 yards: what is 

 the chance that the actual range shall exceed 2000 yards? 

 The ultimately uniform (or random) distribution here is 

 commonly assumed to apply to the various directions in 

 which the gun can be pointed; all possible directions above 



1 See the problem paper of Jan. 18, 1854, in the Cambridge Mathema 

 tical Tripos. 



