BAYES. 299 



respect to x^ we shall find that the terms cancel so as to leave 

 only ^u^, 



551. The general theory of the estimation of the probabilities 

 of causes from observed events was first given by Laplace in the 

 Memoires ...par divers 8avans, Vol. vi. 1774. One of Laplace's 

 results is that if an event has happened p times and failed q 

 times, the probability that it will happen at the next trial is 



J 



x''^' (1 - xY dx 



f 



J 



x'' (1 - xY dx 



Lubbock and Drinkwater think that Bayes, or perhaps rather 

 Price, confounded the probability given by Bayes's theorem with 

 the probability given by the result just taken from Laplace ; see 

 Lubbock and Drinkwater, page 48. But it appears to me that 

 Price understood correctly what Bayes's theorem really expressed. 

 Price's first example is that in which p = 1, and ^ = 0. Price says 

 that "there would be odds of three to one for somewhat more 

 than an even chance that it would happen on a second trial." 

 His demonstration is then given ; it amounts to this : 



/: 



af{l-xYdx .^ 



I x^Q.- xy dx 



J 



i' 



8 



where p = l and q = 0. Thus there is a probability - that the 



chance of the event lies between ^ and 1, that is a probability 



3 



7 that the event is more likely to happen than not. 



552. It must be observed with respect to the result in Art. 549, 

 that in Bayes's own problem we hnonj that a priori any position 

 of ^jF between AB and CD is equally likely ; or at least we know 

 what amount of assumption is involved in this supposition. In 

 the applications which have been made of Bayes's theorem, and 

 of such results as that which we have taken from Laplace in 



