600 PROFESSOR BOOLE ON THE COMBINATION 



6. I define the mathematical probability of an event as the ratio which the 

 number of distinct cases or hypotheses favourable to that event bears to the 

 whole number of distinct cases possible, supposing that to none of those cases 

 the mind is entitled to give any preference over any other. Fundamentally, this 

 definition agrees with that of Laplace. " La theorie des hazards consiste," he 

 remarks, " a reduire tous les evenements du meme genre a un certain nombre de 

 cas egalement possibles c'est a dire tels que nous soyons egalement indecis sur leur 

 existence et a determiner le nombre de cas favorables a l'evenement dont on 

 cherche la probabilite. Le rapport de ce nombre a celui de tous les cas possibles 

 est la mesure de cette probabilite." — Essai Philosophique sur les Probabilites. 



It is implied in this definition, that probability is relative to our actual state 

 of information, and varies with that information. Of this principle Laplace 

 gives the following illustration: — " Let there be three urns, A, B, C, of which we 

 are only informed that one contains black and the other white balls ; then, a ball 

 being drawn from C, required the probability that the ball is black. As Ave are 

 ignorant which of the urns contains black balls, so that we have no reason to 

 suppose it to be the urn C rather than the urn A or the urn B, these three hy- 

 potheses will appear equally worthy of credit, but as the first of the three hypo- 

 theses alone is favourable to the drawing of a black ball from C, the probability 

 of that event is ^. Suppose now that, in addition to the previous data, it is 

 known that the urn A contains only white balls, then our state of indecision has 

 reference only to the urns B and C, and the probability that a ball drawn from 

 C will be black is ^. Lastly, if we are assured that both A and B contain white balls 

 only, the probability that a black ball will issue from C rises into certitude." — 

 Essai Philosophique sur les Probabilites, p. 9. — {Phil. Mag., p. 433.) Our estimate 

 of the probability of an event varies not absolutely with the circumstances which 

 actually affect its occurrence, but with our knowledge of those circumstances. 



7. When the probabilities of simple events constitute our only data, we can, 

 by virtue of the above definition, determine the probability of any logical combi- 

 nation of those events, and this either, 1st, absolutely ; or, 2dly, conditionally. 

 The reason why we can, in this case, more immediately apply the definition is, 

 that not only is no connection expressed among the events whose probabilities are 

 given, but none is implied, nor is any restraint imposed upon their possible com- 

 binations. This, as we shall see, is not the case when the data are the probabi- 

 lities of compound events. 



As an example, let us suppose that the probability of the conjunction of two 

 events, x and y, is required, the data being simply that the probability of the 

 event x is p, and that of the event y is q. Or, to express the problem in a form 

 which we shall hereafter generally employ : 



Given Prob . x =p, Prob. y = q, 



Required Prob. xy. 



