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XXXVII. Note on a Question in the Theory of Probabilities. 

 By A. Cayley*. 



THE following question was suggested to me, either by some 

 of Prof. Boole's memoirs on the subject of probabilities, 

 or in conversation with him, I forget which ; it seems to me a 

 good instance of the class of questions to which it belongs. 



Given the probability a that a cause A will act, and the pro- 

 bability^ that A acting the effect will happen; also the pro- 

 bability that a cause B will act, and the probability q that B 

 acting the effect will happen ; required the total probability of 

 the effect. 



As an instance of the precise case contemplated, take the fol- 

 lowing : say a day is called windy if there is at least w of wind, 

 and a day is called rainy if there is at least r of rain, and a day 

 is called stormy if there is at least W of wind, or if there is at 

 least R of rain. The day may therefore be stormy because of 

 there being at least W of wind, or because of there being at least 

 R of rain, or on both accounts ; but if there is less than W of 

 wind and less than R of rain, the day will not be stormy. Then 

 a is the probability that a day chosen at random will be windy, 

 p the probability that a windy day chosen at random will be 

 stormy, ft the probability that a day chosen at random will be 

 rainy, q the probability that a rainy day chosen at random wiJl 

 be stormy. The quantities X, p, introduced in the solution of 

 the question mean in this particular instance, X the probability 

 that a windy day chosen at random will be stormy by reason of 

 the quantity of wind, or in other words, that there will be at 

 least W of wind, fi the probability that a rainy day chosen at 

 random will be stormy by reason of the quantity of rain, or in 

 other words, that there will be at least R of rain. 



The sense of the terms being clearly understood, the problem 

 presents of course no difficulty. Let X be the probability that 

 the cause A acting will act efficaciously ; yu, the probability that 

 the cause B acting will act efficaciously -, then 

 p=X+(l-X)n/3 

 q = fi+(l—p)oiX, 

 which determine X, //, ; and the total probability p of the effect 

 is given by p = Xct+pp-\/i*l3, 



suppose, for instance, a=l, then 



p = X-\-i—Xfift, q = p, + X—Xp,, p^zX+fjbfi—'kfif}, 

 that is, p=p, for p is in this case the probability that (acting 

 a cause which is certain to act) the effect will happen, or what 

 is the same thing, p is the probability that the effect will happen. 



Machynlleth, August 16, 1853. 



* Communicated by the Author. 

 S2 



