ROBERTS. 53 



and r Chances for c, then my hazard is worth ^J- — — " Our 



]_)^- q + r 



author demonstrates this, and intimates that it may be extended 



to the case when there are also s Chances for d, &c. 



Our author then considers the game of Hazard. He gives an 

 investigation similar to that in De Moivre, and leading to the 

 same results; see Doctrine of Chances, page IGO. 



87. The first part of the book concludes thus : 



All those Problems suppose Chances, which are in an equal proba- 

 bility to happen; if it should be suppos'd otherwise, there will arise 

 variety of Cases of a quite different nature, which, perhaps, 'twere not 

 unpleasant to consider : I sliall add one Problem of that kind, leaving 

 the Solution to those who think it merits their pains. 



In Parallel ipipedo cujus latera sunt ad iuvicem in ratione a,b,c: 

 Invenire quota vice quivis suscipere potest, ut datum quodvis planum, 

 v.g. aSjaciat. 



The problem was aftersvards discussed by Thomas Simpson ; it 

 is Problem xxvil, of his Nature and Laius of CJiance. 



88. It will be convenient to postpone an account of the second 

 part of the book until after we have examined the works of De 

 Moivre. 



89. We next notice An Arithmetical Paradox, concerning the 

 Chances of Lotteries, by the Honourable Francis Roberts, Esq. ; 

 Fellow of the R S. 



This is published in Vol. xvii. of the Philosophical Trans- 

 actions, 1693 ; it occupies pages 677 — 681. 



Suppose in one lottery that there are three blanks, and three 

 prizes each of 16 pence ; suppose in another lottery that there are 

 four blanks, and two prizes each of 2 shilliugs. Now for one 

 drawing, in the first lottery the expectation is ^ of 16 pence, and in 

 the second it is J of 2 shillings ; so that it is 8 pence in each case. 

 The paradox which Roberts finds is this ; suppose that a gamester 

 pays a shilling for the chance in one of these lotteries ; then 

 although, as we have just seen, the expectations are equal, yet the 

 odds against him are 3 to 1 in the first lottery, and only 2 to 1 in 

 the second. 



