the Theory of Probabilities. 31 



l-w=(l-«)(l-/3), 

 .-. «=l-(l-«)(l-/3), 

 a result obviously correct. For if either cause necessarily pro- 

 duces the eflect, the probability of the effect is equal to the pro- 

 bability that both the causes will not be absent. 



Again, suppose that « and /S are very small, so that their 

 product may be neglected, then the solution gives 



whence we find 



u = a.p+/3q. 



Now this is the known form of the solution when the causes 

 are mutually exclusive, so that they cannot coincide. But the 

 smaller their separate probabilities, the smaller is the probability 

 of their coincidence. And in the limit the two probabilities 

 correspond. 



I suppose that few persons, after reading Mr. Cayley's solu- 

 tion and noticing the above verifications, would feel any doubt of 

 its correctness; and yet it is certainly erroneous. Take the 

 particular cases of j3 = l and 5=0. It is evident that the pro- 

 bability of the effect ought then to be a. If the cause A always 

 produces the effect, and the cause B never, the probability of the 

 effect, there being no other causes to which it can be ascribed, 

 will be equal to that of the cause A. The equation (4), howevei', 

 becomes on clearing of fractious, 



(l-^^)(l-^-«) = (l-«)(l-/3)(l-^^), 

 whence either w = l, or 



l-fi-u={l-u){l-^), 

 wherefore 



« = (l-/3)-(l- )(1-^) 



=«(l-/3). 

 But neither of these results is correct. 



2nd. I will now exhibit the true solution of the problem. 

 Representing, as befoi-e, by u the probability of the effect, the 

 value of that quantity will be determined by the solution of the 

 quadratic equation 



1— « • up+^q—u 



Of this equation that root must be taken which is at the same 

 time not less than each of the two quantities 



up and /3y, (6) 



