4i0 BICQUILLEY. 



III. Suppose that P and Q happen together tivice. Tlie cor- 

 responding chance is 



r" ..w._f? _n_5 42,.u.-m-n + 2 



[2 I m - 2 I ?i-2 I ^ - 9?? - ?2 + 2 

 And so on. 



819. As another example of the hind of problem noticed in 

 the preceding Article, we may require the cliance that in \l trials P 

 and Q shall each happen at least once. The required chance is 



1 _ (1 _^; _ ^)^ _ (1 _ ^ _ Q/^ + (1 -^ - 5^ - ty. 



See also Algehra, Chapter LVI. 



820. Pages 111 — 133 contain the solution of some examples. 

 Two of them are borrowed from Buffon, namely those which we 

 have noticed in Art. 649, and in the beginning of Art. 650. 



One of Bicquilley's examples may be given. Suppose p and q 

 to denote respectively the chances of the happening and failing of 

 an event in a single trial. A pla3^er lays a wager of a to & that the 

 event will happen ; if the event does not happen he repeats the 

 wager, making the stakes ra to rh ; if the event fails again he 

 repeats the wager, making the stakes r'^a to r^5 ; and so on. If the 

 player is allowed to do this for a series of n games, required his 

 advantage or disadvantage. 



The player's disadvantage is 



This is easily shewn. For qa ~])h is obviously the player's dis- 

 advantage at the first trial. Suppose the event fails at the first 

 trial, of which the chance is q ; then the wager is renewed ; and 

 the disadvantage for that trial is qar — ph\ Similarly (f is the 

 chance that the event will fail twice in succession ; then the wager 

 is renewed, and the disadvantage is qar^—pbr^. And so on. If 

 then qa is greater than ph the disadvantage is j)ositive and in- 

 creases with the number of games. 



Bicquilley takes the particular case in which a = 1, and 



5 + 1 . 



^ = — -J — ; his solution is less simple than that which we have 



