360 Mr. A. Cayley on a Question in the 



or substituting for u its value. 



a 



x=0, s=l, j=a 



V 



6' p' + Pq-a* 

 fy'tf' + Pq) t _q(/3' + /3q~a) 



(l-(3q) {l 3> + l3q)-a{3» if ffP+0q) ' 



which give x, s, y, t. It seems difficult to interpret the equa- 

 tions X = 0y 5=1. 



An intermediate system of equations, putting therein # = 0, 

 5=1, and therefore #' = 1, s' = 0, is 



Jy_ _s' i!y ' J _ ij} H ty 



u — u u—/3q u fi' + fiq — u cc + ftq—u 



x t 



which in fact lead to the foregoing values of u, x, s, -„ y, and -,• 



s z 



The probabilities of the compound events AB, &c. are in 



general as follows : viz. 



Probs. of 



AB :: to zy{st + sV), 



AB' „ xijt ! , 

 A'B „ x's'y, 



A'B f „ My't'. 



Or in the present case, dividing by s' and then writing #=0, 5 = 1, 



Probs. of 

 AB :: to %yt, 





AB' „ p'f. 



A'B „ y, 

 A'B' „ y'l'. 

 In particular, if p = l, q = 0, then 



x rt 



X — 0) 5=1, -7 = ^ 



s' l-u-p 



y=^r u > p=0; thatis,^ 1 ^^, / = 0, *' = 1. 

 So that the probabilities of the four events are as 



* ■ £ . 1 ~ a " /3 . 

 1-a'l-a' l-« ' 



that is, the probabilities of AB, AB', A'B, A'B' are 



0> *> ft l-«-/3 



