C II A 



717 



C H A 



fly proceeding ai in the foregoing problems, it 

 will Hppear, that when 



then one term 

 = 4 Toft' 

 == 9 i then 'J terms 

 = l()f are 



then 3 terms 

 are 



= '21 : 

 = '2'J 



then 4- terms 

 are 





So that, in general, when 



I <hen.r.f 1 f greater J than , , ? . 

 3 terms are lcss j 

 Therefore, when m=6x-f 4, there will be some- 

 thing more than an equal chance of the required ef- 

 fect's happening x-{- 1 times. And since m=r6jr-f-^ 



w 4 



therefore =. Also, since r=o:4-l, therefore 



.T=r 1 ; hence ! = 



<; 



, wz -f-2 

 and r= = . 



Hence it appears, that in a lottery in which there 

 are five blanks to one prize, if m tickets are bought, 



- - prizes may 



then on an equality of chance, 

 expected, 



be 



Cor. Since r 1 = 



in 4- 

 ~6~' 



therefore /=6r 2. 



Whence it appears, that if it be required to know 

 how many tickets should be purchased in such a lot- 

 tery, in order, on an equality of chance, to expect r 

 prizes ; the answer will be 6> 2. Therefore, in or- 

 der to have 1, 2, 3, 4, 5, &c. prizes, there should be 

 purchased 4, 10, 16, 22, 28, &c. tickets. 



PROBLEM XXXIII. 



4-3. In a lottery where the number of blanks is to 

 the number of prizes as b to a ; how many tickets 

 must be purchased to procure an equal chance for p 

 or more prizes ? 



SOLUTION. From a careful observation of the co- 

 rollaries to the five preceding problems, it will ap- 

 pear, that the series which in each expresses the num- 

 ber of tickets that ought to be purchased, in order 

 to procure an equality of chance for the having 1, 2, 

 M, 4-, 5, &c. prizes, do severally differ by 2, 3, 4, 5, 

 <), which, in each separate question, is the number of 

 blanks -f-1, or (since there is supposed only one 

 prize to a certain number of blanks) the number of 

 chances which one ticket has of being either a blauk 

 or a prize. Thus in corollary to Prob. 



28 i 



bs 



io5 



: c " 



U 



o 



*8^ 



C C i 



o 



o c 



^! .S rt S 



g- .s 



'3 o p,p,f 



1, 3, 5, 7, 9, &c. 



2, 5, 8,Il,H,&c, 



3, 7,11,15,19, &c 

 4-, 9, 14,19,24-, &c 

 4, 10, 16, 22,28, &c, 



j)'ja jj 



^3 



CO t< 



5^ w ** 



p ^ i. 



So o 



6 -5.3 



fore we may conclude, iltat the tatr.e 

 will happen in all succeeding questions of this 

 and consequently, that if the brat term of the tfriet 

 can be obtained, then all the rest will be found bjr 

 the addition of (a-\-b). 



Now the first term of this series may be found br 

 Prob. X\VII. where the number of tickets which 

 must be purchased, that the buyer may have an equal 



. . . log. 2. 



chance to have one pn/c, is > - 7 pW r - r- 



log. (fl+6) tog. 6 



Therefore this quotient, if an integer, or the neit 

 greater integer if a fraction, will be the first term of 

 the series. And if we call this quotient </, and put 

 a -f-4=5, then, in order to have an equal chance for 

 1, 2, 3, 4-, 5 prizes, we must purchase g, o+*, 9+2*. 

 7^-3*, 7 + 4*, &c. tickets. Or universally, in order 

 to have an equal chance for p prizes, we must pur- 

 chase 7 4- (p-\ )* tickets. 



For the application of the doctrine of chances to 

 aanuities, see ANNUITIES. 



List of writers and works en the subject of chin- 

 ces. 



Pascal and Fermal. 



Jac. Bernoulli Ars Conjectandi Opus PostKumum. 



Letire d une Amy, sur les parties du Jeu depaume; 

 subjoined to the Ars Conjectandi. 



De ratiodniis in Litdo Alea t Auctore Christ. Hu- 

 genio. 



Essai d* analyse sur lesjeux de hasard, par Mtfn- 

 mort. 



De Moivre's Doctrine of Chances. 



Miscellanea Ana/ytica, by the same author. 



Specimina artis cotijcctandis ad questioner juris ap* 

 plicate, by Nicolas Bernoulli, in the Leipsic Acts, 

 1711. 



Simpson On the Nature and Lams of Chance. 



Sur I* usage du principe de la raison sujffi sante dan s 

 le calcul des probability par Beguelin. Me moires de 

 Berlin, 



i 

 i 



Opuscules Mathematiques ; par D'Alembert. 



Essai xnr I' application de V analyse a la probability 

 dcs decisions <) la pluralitc des voix, par Coodorcet. 



Dodson's Mattiematical Repository. 



Emerson On the Lans of Chance, in his Miscella- 

 nies. () 



CHANTRY, or CIIAUNTRV, (Cantaria,} a small 

 chapel or church, or private altar in a cathedral or 

 other public place of worship, with an endowment 

 for one or more priests, on condition that they should 

 sing mass, and perform other divine services for the 

 soul of the founder, or of such also of his descendants 

 or other relations as he may have provided for by the 

 grant. These endowments, however, as originating 

 in the Romish superstition, were effectually abolished 

 by Stat. 1 Ed. VI, c. 14, which declares all entry 

 into the lands, or other revenues, in terms of the foun- 

 dation, unlawful, and confers the property upon the 

 king, under certain exceptions in favour of universi- 

 ties and other public seminaries. From Dugdale's 

 history of St Paul's, it appears that not fewer than 

 forty-seven chantries belonged to that church, (j. B. ) 



CHAPEL (Capella,) a place of divine worship, 

 and of which there are commonly reckoned four dif- 

 ferent sorts. 1st, Chapels of case, which are provi- 

 ded for the ease or convenience of the parishiopej-j, 



