602 PROFESSOR BOOLE ON THE COMBINATION 



means of the simple events x, y, z, &c, in accordance with the notation of the 

 calculus of Logic. 



By the above theorem the probability of any compound event is determined 

 absolutely, when the probabilities of its simple components are given. 



9. And by the same mode of investigation, the probability of any combination 

 may be determined conditionally, i.e., the probability which the combination will 

 have under a given condition consisting in the happening of some other combina- 

 tion. Thus, if our data are as before, 



Prob. x=p, Prob. y — q, Prob. z = r, &e. 



and if we require the probability that if the event <p (x, y, z . .) present itself, the 

 event -^ (x, y,z . .) will be present at the same time, Ave may demonstrate the fol- 

 lowing result, viz. : — 



Prob. that if $ (x, y,z . .) happen, 4, (x, y,z . .) will be present also 



_ X (P> 9>r • •) (2 , 



{p, q, r . .) ^ ' 



where the form of the function ^ is determined by multiplying together, ac- 

 cording to the p rinciples of the calculus of Logic, the functions^, y,z . .) and 

 sj, (x, y, z . .), and representing the result by x {*> y> z ■ •)• — {Laws of Thought, p. 258, 

 Prop. I.) 



10. I postulate that when the data are not the probabilities of simple events, 

 we must, in order to apply them to the calculation of probability, regard them, 

 not as primary, but as derived from some anterior hypothesis, which presents the 

 probabilities of simple events as its system of data, and exhibits our actual data 

 as flowing out of that system, in accordance with those principles which have 

 already been shown to be involved in the very definition of probability. 



The ground of this postulate is, that to begin with the simple and proceed to 

 the complex, seems to be, in all questions involving combinations such as we are 

 here concerned with, a necessary procedure of the understanding. The calcula- 

 tion of probability depends upon combinations subject to a peculiar condition, 

 viz., that they shall always present to us a series of cases or hypotheses, to none 

 of which the mind is entitled to attach any preference over any other. AVe can- 

 not, in endeavouring to ascend from the complex to the simple, secure the main- 

 tenance of this condition ; but we can do so in descending from the simple to the 

 complex. We have had an illustration of this truth in the reasoning by which 

 we deduced the expression for the probability of the complex event xy from the 

 probabilities of the simple events x and y, supposed to be given. And the me- 

 thods which have been actually employed in the solution of problems whose im- 

 mediate data were not the probabilities of simple events, have in fact rested upon 

 the postulate above referred to. Thus in questions relating to juries, the imme- 

 diate data are the probabilities, founded upon continued observation, that a de- 

 cision will be unanimous, or that it will be pronounced by a given majority, &c. 



