296 BATES. 



Bayes gives the principle by which we must calculate the 

 probability of a compound event. 



Suppose we denote the probability of the compound event by 



p 



-^y the probability of the first event by z, and the probability 



of the second on the supposition of the happening of the first 



7 P h 



by -^ . Then our principle gives us ^^T = ^ '^Jf> ^^^ therefore 



p 



z = — . This result Bayes seems to present as something new 







and remarkable ; he arrives at it by a strange process, and enun- 

 ciates it as his Proposition 5 in these obscure terms : 



If there be two subsequent events, the probability of the 2nd -^ 



P . . 



and the probability of both together -^, and it being 1st discovered 



that the 2nd event has happened, from hence I guess that the 1st event 



. . P 

 has also happened, the probability I am in the right is -r-. 



Price himself gives a note which shews a clearer appreciation 

 of the proposition than Bayes had. 



b^o. We pass on now to the remarkable part of the essay. 



Imagine a rectangular billiard table ABCD. Let a ball be rolled on 



it at random, and when the ball comes to rest let its perpendicular 



distance from A She measured ; denote this by x. Let a denote the 



distance between AB and CD. Then the probability that the 



. c — h 



value of X lies between two assio^ned values J and c is . This 



a 



we should assume as obvious ; Bayes, however, demonstrates it 



very elaborately. 



54<6. Suppose that a ball is rolled in the manner just ex- 

 plained ; through the point at which it comes to rest let a line EF 

 be drawn parallel to AB, so that the billiard table is divided into 

 the two portions AEFB and EDCF. A second ball is to be rolled 

 on the table ; required the probability that it will rest within the 



