Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/292

 

CHAPTER XVIII.

ELEMENTARY ILLUSTRATIONS OF THE GENERAL METHOD IN PROBABILITIES.

1. IT is designed here to illustrate, by elementary examples, the general method demonstrated in the last chapter. The examples chosen will be chiefly such as, from their simplicity, permit a ready verification of the solutions obtained. But some intimations will appear of a higher class of problems, hereafter to be more fully considered, the analysis of which would be incomplete without the aid of a distinct method determining the necessary conditions among their data, in order that they may represent a possible experience, and assigning the corresponding limits of the final solutions. The fuller consideration of that method, and of its applications, is reserved for the next chapter.

2. Ex. 1.—The probability that it thunders upon a given day is $\scriptstyle {p}$, the probability that it both thunders and hails is $\scriptstyle {q}$, but of the connexion of the two phenomena of thunder and hail, nothing further is supposed to be known. Required the probability that it hails on the proposed day.
Let $\scriptstyle {x}$ represent the event—It thunders.
Let $\scriptstyle {y}$ represent the event—It hails.
Then $\scriptstyle {xy}$ will represent the event—It thunders and hails; and the data of the problem are Prob.$\scriptstyle {x=p}$, Prob.$\scriptstyle {xy=q}$. There being here but one compound event $\scriptstyle {xy}$ involved, assume, according to the rule, (1)$\scriptstyle {xy=u}$. Our data then become (2)Prob.$\scriptstyle {x=p}$, Prob.$\scriptstyle {u=q}$; and it is required to find Prob.$\scriptstyle {y}$. Now (1) gives