BATES. 297 



space AEFB. If x denote the distance between AB and ^i^tlie 



X 



required probability is - : this follows from the preceding Article. 



547. Bayes now considers the following compound event : 

 The first ball is to be rolled once, and so EF determined ; then 

 p +q^ trials are to be made in succession with the second ball : 

 required the probability, before the first ball is rolled, that the 

 distance of EF from AB will lie between h and c, and that the 

 second ball will rest p times within the space AEFB, and q times 

 without that space. 



We should proceed thus in the solution : The chance that EF 



falls at a distance x from AB is — ; the chance that the second 



a 



event then happens p times and fails q^ times is 



hence the chance of the occurrence of the two contino^encies is 



a \p_\q_ \«/ \ «/ * 

 Therefore the whole probability required is 



a\p[q 





(i-I)''^- 



Bayes's method of solution is of course very different from the 

 above. With him an area takes the place of the integral, and 

 he establishes the result by a rigorous demonstration of the ex 

 ahsurdo kind. 



548. As a corollary Bayes gives the following: The proba- 

 bility, before the first ball is rolled, that EF will lie between AB 

 and CD, and that the second event will happen p times and fail q 

 times, is found by putting the limits and a instead of h and c. 

 But it is certain that EF will lie between AB and CD. Hence we 



