SIMPSON. 207 



The result in modern notation is a fraction of which the nume- 

 rator is 



\a \h [c 



X "; ; :; X X . • . j 



\p \<^-p \q \'b-q 



r 



c — r 



and the denominator is : — —•- 



\m\n — m 



where ?z = a + & + c+... 



This is apparently the problem which Simpson describes in his 

 title page as '^A new and comprehensive Problem of great Use in 

 discovering the Advantage or Loss in Lotteries, Raffles, &c." 



861. Simpson's Problem x. relates to the game of Bowls ; see 

 Art. 177. Simpson gives a Table containing results for the case of 

 an indefinitely large number of players on each side, but he does 

 not fully explain his Table ; a better account of it will be found in 

 Samuel Clark's Laws of Chance, pages 63 — 65. 



S62. Simpson's Problem XV. is to find in how many trials one 

 may undertake to have an equal chance for an event to occur r 

 times, its chance at a single trial being known. Simpson claims 

 to have solved this problem "in a more general manner than 

 hitherto ;" but it does not seem to me that what he has added to 

 De Moivre's result is of any importance. We will however give 

 Simpson's addition. Suppose we require the event to happen 



r times, the chance for it in a single trial being j. Let 



2' = - ; and suppose that q^ is large. Then De Moivre shews that 



in order to have an even chance that the event shall occur r times 



we must make about q ( ^ ~ tt; ) trials ; see Ai't. 262. But if ^ = 1 



the required number of trials is exactly 2r — 1. Simpson then 



proposes to take as a universal formula 2'(^'~t7v)+^~t^j this 



is accurate when g[ = l, and extremely near the truth when q is 

 large. 



