Mr. J. M. Wilson on some Problems in Chances. 171 



lines are drawn at random, as the fourth line is, it is equally 

 likely that all of the triangles should be infinite. 



Let the four lines be drawn in any manner ; call the lines 



a, b, c, d; each of them in succession may be considered the 

 fourth line; and it will of course happen that in two of the four 



cases the fourth line does intersect the triangle formed by the 

 first three ; and therefore the chance required is J. 



In this solution the italicized part contains the assumption, to 

 which some attention should be given. It in fact substitutes 

 simultaneous for successive drawings of the lines in the figure. 

 Viewed in this light I do not think it can be objected to; for by 

 hypothesis all the lines are at random and independent of one 

 another; and this once granted, the rest of the solution presents 

 no kind of difficulty. 



It may be viewed as a compendious way of presenting the fol- 

 lowing solution. Let the lines be drawn in the order a, b, c, d. 

 Among the infinite series of figures so obtained, it will necessarily 

 follow that d and a, d and b, dand c will sooner or later change 

 places, the others remaining where they were. Hence it is clear 

 that the figures, if every possible figure could be conceived as 

 drawn, could be arranged in groups of four, the figures being the 

 same in each group, and d occupying each of the four positions 

 of the lines in succession ; that is, in each group d will twice in- 

 tersect the triangle formed by a, b, c } and twice it will not ; or 

 the chance required is, as before, J. 



It must be observed that wherever the first method of solu- 

 tion is applicable, it may be looked on as a succinct way of sta- 

 ting this kind of proof, if this latter is thought more satisfactory. 

 The next problem is the " four-point problem." 



If four points be taken at random in a plane, what is the 

 chance that one of them lies inside the triangle formed by the 

 other three. 



(1) A point at random may be viewed as the intersection o 

 two lines at random. 



(2) Four random lines determine three sets of four random 

 points. 



(3) In one only of these three sets is one of the four within the 

 triangle formed by the other three. 



(4) And therefore the chance required is -j. 



With respect to (2), it may be remarked that if the lines are a, 



b, c, d, and ab represent the point of intersection of a and b, 

 the three sets of four points are 



ab, ac, db, dc; 



ba, be, da, dc; 



ca, cb, da, db. 



