2QQ on the DOCTIIINES OF CHANCE. 



is not to be fouiid in de Moi?re, or I may add iw any other 

 author that I have seen on the La.ws of Chancre, peihaps to 

 say any thing farther on the subject may be deemed unne- 

 cessary. 



Lest however the^doubts of Opsimath should not be fully 



_ , ,„,^ removed, let as proceed with liim a little farther. — He ooes 

 FaiftheF state- ' r o 



nient f)£ on to say, or rather to quote, that " any one undertaking to 



duubis, a ^.j^g^ jj^ g(,g jj^ j.^yQ throws of one die, has for the first proba- 



*' bility I" , as proved : should the first fail, then the second 

 " remains, which is ^ likewise; but the chance of the rirnt 

 *' failing is |, as that of its succeeding is j^; therefore the se- 

 " cond throv/ has only -^ of A for its chance of success, which 

 " added to the chance of casting an ace the first throw, is i-1-^ 

 *' of |-— ^i-; the first throw being -/'-, the second only ^rV'" 

 " This doctrine," Opsimath adds, " 1 cannot grant" — " be- 

 cause" »ays he, " nothing can prevent him of the second 

 throw, except his succeeding in the first." Very true, but 

 Answet to ^''^ succeeding in the first mai/ and certainly iclU prevent 

 xiicsc. him of the ^eco7icZ throw. Opsimath should recollect, that 



the probability of the event's happening is calculated before 

 either throw is made; and that, till the Jirst throw is made, 

 it is uncertain whether the second will be required ; and 

 consequently, that, tliough the second throw has " the full 

 " force and virtue of -^ chance" after the first is over, yet, be- 

 fore that event, its value can only be i multiplied by the 

 probability that the first throw w'lWfail, for on the failure of 

 the first depends the necessity of the second — that is, since 

 ibe probability of the c.v?5/e??ce of the second throw, if I may 

 so term it, is, before the first takes place, only ~; and, should 

 it exist, the probability of its producing an ace is only J ; 

 therefore, before either throw is made, the value of the pro- 

 baliility of the second is only | of ^, that is ^^g-, which, ad- 

 ded to ^, the probability lor the first throw, gives -J | for the 

 probability on the two. 



« , w IShould tiiis consideration of the dependence of the second 



Answered by •' 



deUuciiivg the ever.t upon the first fail to remove the scruples of Opsimath, 

 fal^me '^'' ^^ y*-'^' ^ thiiik there will be no difficulty in convincing him 

 upon the principles, which he has himself admitted, that iv 

 express the true probability of casting an ace once in tv\o 

 thrywi. Since the probability of an event's happening, to- 

 gether 



