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



So in what is called the &quot; three-point problem &quot; : Three 

 points in space are selected at random ; find the chance of 

 their forming an acute-angled triangle. What is done is to 

 start with a closed volume, say a sphere, from its superior 

 simplicity, find the chance (on the assumption of uniform 

 distribution within this volume) ; and then conceive the con 

 tinual enlargement without limit of this sphere. So regarded 

 the problem is perfectly consistent and intelligible, though I 

 fail to see why it should be termed a random selection in 

 space rather than in a sphere. Of course if we started with 

 a different volume, say a cube, we should get a different 

 result; and it is therefore contended (e.g. by Mr Crofton in 

 the Educational Times, as already referred to) that infinite 

 space is more naturally and appropriately regarded as tended 

 towards by the enlargement of a sphere than by that of a 

 cube or any other figure. 



Again : A group of integers is taken at random ; show 

 that the number thus taken is more likely to be odd than 

 even. What we do in answering this is to start with any 

 finite number n, and show that of all the possible com 

 binations which can be made within this range there are 

 more odd than even. Since this is true irrespective of the 

 magnitude of n, we are apt to speak as if we could conceive 

 the selection being made at random from the true infinity 

 contemplated in numeration. 



8. Where these conditions cannot be secured then it 

 seems to me that the attempt to assign any finite value to 

 the probability fails. For instance, in the following problem, 

 proposed by Mr J. M. Wilson, &quot;Three straight lines are 

 drawn at random on an infinite plane, and a fourth line is 

 drawn at random to intersect them : find the probability of 

 its passing through the triangle formed by the other three &quot; 

 (Ed. Times, Reprint, Vol. v. p. 82), he offers the following 



