Page:Mind (New Series) Volume 9.djvu/92

 78 HUGH MACCOLL : AF, AF, Hence, in the notation of chances, _ 1 j^_ + .2 ^_ + ^3. fi ) by But F, = _F_ 2 = Fg _ 1 " 3' Therefore, A = 1 / A. A A S^F; F 2 F 3 ; = chance that A is Similarly, A _ e = 1 f A _ c + A ! + A I = i (1 + + 2 0) = | = chance that A is certain ; And _Ai = l/Ai + AiAi T 3 I W V F~ J B  *1 r 2 r 3 ' = i(0+l + 0) = = chance that A is impossible.  ls< ' J^ -. A^ 11 f = chance that A is possible. Thus, from the data of the three preceding random figures > we have found - = -~, an equational statement which implies A 11 ' 9 . But it is clear that from different data we might have arrived at a different result. Let us, for example, suppose Fig. 2 to be similar to either Fig. 1 or to Fig. 3. Then, instead of our former result, we shall have

and, therefore, Ai' _ ..

That is, instead of A* 1 '", we shall have A 11 ". Let us now suppose that we have three collections of figures with corresponding data, namely, 1 Kj, K 2, K 3 ; of which Kj and K 2 lead each to this last conclusion Ai", while K 3 leads to the conclusion A" 1 '", the denial of Ai". Out of the three * collections K 1; K 2, K 3 (which we assume to be the only 1 Each of the three collections, K,, K 2 , K 3 , is understood to consist of two, three, four or any number of figures.