BISHOP TERROT ON PROBABILITIES. 375 



Probability, as Mr Boole in his Laws of Thought, properly defines it, is " Ex- 

 pectation founded upon partial knowledge." Events, therefore, of which we 

 possess complete knowledge, and events of which we possess no knowledge, are 

 equally, by the terms of the definition, excluded from the class of probable events, 

 that is to say, of events to which the calculus of probabilities can be applied. If 

 we are certain that an event has happened, we totally neglect and are unaffected 

 by any subsequent information, which, but for that certainty, would have given 

 to the event a definite probability expressed by a proper fraction ; and never 

 think of looking for a form by which to combine such fraction with the unit ex- 

 pressing the certainty. If, again, we derive from experience or observation a 

 definite probability of any event, such, for example, as the probability for drawing 

 a white ball from an urn, whose contents are given ; namely, the fraction whose 

 numerator is the number of white balls, and its denominator the total number of 



balls contained, we never think of combining this with the ~ which is assumed 



as the probability when nothing whatever was known, except that the ball drawn 

 must be either white or not white. Complete knowledge comprehends all pre- 

 vious partial knowledge ; and, therefore, all fractional expressions for probability 

 derived from the latter, are virtually contained in the unit, which is the expres- 

 sion adopted for the certainty produced by the former. On the other hand, 

 partial knowledge destroys total ignorance, and any inference that may be drawn 

 from it. It comprehends the hypothesis that the event may, and that it may not 

 happen, with a definite probability to each, which do not supplement but super- 

 sede the probabilities of -x previously assumed for each. I cannot conclude without 



suggesting a doubt, whether ^ be at any time the proper expression for the pro- 

 bability of an event which is " neither likely nor unlikely in regard of evidence." 

 It seems more analogous with the practice in other cases to express such pro- 

 bability by the indefinite fraction -• If this expression be applied to either of the 



probabilities constituting the compound probability - — , the compound proba- 



Q -L /y* w* 



bility will be reduced to the remaining simple probability, for ^— =-. And this 



agrees with the necessary action of the mind, which takes no note of its original 

 ignorance, after it has arrived at a definite probability from partial knowledge. 



(12.) Hitherto, I have been speaking of the combined result of two probabilities 

 of the same event, derived from distinct sources of partial knowledge; and I have 

 shown that to obtain a definite result, the mere ratio in such case is insufficient, 

 and that the actual number of favourable and unfavourable cases in each of the 

 data is requisite. 



