162 AN EXAMINATION OF SOME QUESTIONS 



In the instance we are now considering, if the number of 

 successful and of unsuccessful cases are equal, the gamester 

 who adopts this system of play will always win, for in that 

 case p=:q, and W 1 ~pv; as a numerical example of the 

 formula just investigated. Let us suppose 



A person stakes 100 shillings on the event of a piece of mo- 

 ney thrown into the air falling with one of its faces uppermost, 

 in preference to the other ; whenever he wins he diminishes 

 his stake five shillings, and whenever he loses he increases it 

 by the same sum. Supposing he makes 800 successive bets, 

 and wins p, and loses q of them. 



If p and q are equal, each being 400, his profit will be 

 400 X 5 = 2000 shillings. 



From this it appears, that he will win even though a smaller 

 number than one-half of the events prove favourable. In or- 

 der to determine how many times he must win, that he may 

 neither lose nor gain on the whole number, we must make the 

 value of W equal to zero, putting p -j- q — a ; this gives for 

 the required value of p 



p= +i [ a +0±l*/£+ ai 



in the present instance, a — 800, u =. 100, v = 5, which 

 gives 



p=± (800 -f 10) + I JlOO + 800 = 405 + 15 ; 



the lower sign being employed, we have p — 390. 



In fact, on substituting this number in the formula (1) we 

 find that he neither wins nor loses money, although the num- 

 ber of unfavourable events has been greater by twenty than the 

 number of favourable ones. The other root of the equation 



points 



