the Theory of Probabilities. 433 



i. e. favourable in the above sense to its not happening, the pro- 



171 



bability of the event is measured by the fraction — . 



It does not, I think, need proof that the principles of the 

 theory of probabilities must be derived either, — 1st, from the 

 nature of probability as set forth in its measure ; or 2ndly, from 

 its connexion with logic and language. Commencing with the 

 former source, I remark that it is implied in the definition that 

 probability is always relative to our actual state of information 

 and varies with that state of information. Laplace illustrates 

 this principle by supposing the following case. Let there be 

 three urns, A, B, C, of which we are only informed that one 

 contains black and the two others white balls ; then, a ball being 

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

 we 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 hypotheses will appear equally 

 worthy of credit ; but as the first of the three hypotheses alone 

 is favourable to the drawing of a black ball from C, the proba- 

 bility of that event is ^. Suppose, now, that in addition to the 

 o 



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.) Here it is seen that our estimate of the probability of an 

 event varies with our knowledge of the circumstances by which it 

 is afiected. In this sense it is that probability may be said to 

 be relative to our actual state of information. 



Let us, in further illustration of this principle, consider the 

 following problem. The probability of an event x is measured 



by the fraction — , that of an event y by the fraction -, but of 



the connexion of the events x and ^ absolutely nothing is known. 

 Required the probability of the event xy, i. e. of the conjunction 

 of the events x and y. 



There are (see definition) a cases in which x happens, to m 

 cases in which it happens or fails; and concerning these cases 

 the mind is in a state of perfect indecision. To no one of them 

 is it entitled to give any preference over any other. There are, 

 in like manner, b cases in which y happens, to n cases in which 

 it happens or fails ; and these cases are in the same sense equally 



