121] 
103 
121. 
NOTE ON A QUESTION IN THE THEORY OF PROBABILITIES. 
[From the Philosophical Magazine, vol. vi. (1853), p. 259.] 
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 probability p that A 
acting the effect will happen; also the probability /3 that a cause B w T ill 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 following: 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, ¡3 the probability that 
a day chosen at random will be rainy, q the probability that a rainy day chosen at 
random will be stormy. The quantities g introduced in the solution of the question 
mean in this particular instance, A, 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; g 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.