Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/311

CHAP. XIX.] CHAPTER XIX. OF STATISTICAL CONDITIONS. Y the term statistical conditions, I mean those conditions which must connect the numerical data of a problem in order that those data may be consistent with each other, and therefore such as statistical observations might actually have furnished. The determination of such conditions constitutes an important problem, the solution of which, to an extent sufficient at least for the requirements of this work, I purpose to undertake in the present chapter, regarding it partly as an independent object of speculation, but partly also as a necessary supplement to the theory of probabilities already in some degree exemplified. The nature of the connexion between the two subjects may be stated as follows: There are innumerable instances, and one of the kind presented itself in the last chapter, An investigation of the laws of thought/Chapter 18, in which the solution of a question in the theory of probabilities is finally dependent upon the solution of an algebraic equation of an elevated degree. In such cases the selection of the proper root must be determined by certain conditions, partly relating to the numerical values assigned in the data, partly to the due limitation of the element required. The discovery of such conditions may sometimes be effected by unaided reasoning. For instance, if there is a probability $$\scriptstyle{p}$$ of the occurrence of an event $\scriptstyle{A}$, and a probability $$\scriptstyle{q}$$ of the concurrence of the said event $\scriptstyle{A}$, and another event $\scriptstyle{B}$, it is evident that we must have But for the general determination of such relations, a distinct method is required, and this we proceed to establish. As derived from actual experience, the probability of any event is the result of a process of approximation. It is the limit of the ratio of the number of cases in which the event is observed to occur, to the whole number of equally possible cases which