66 ESSAY ON PROBABILITIES. 



1.2.3.4x5x3.4.5 Qr 5 



1x1.2.3x8.9.10.11 33 



That all shall be white, or all black (p = 4, q = 4, 

 second and third formulae), the chances are 



Jli,or .land 3 - 4 ' 5 - 6 , or 1 

 8.9.10.11 33 8.9.10.11 22 



and the verification of the whole is 

 73571 



J. 4 1 -I = 1 



22 11 33 33 22 



which must be, since one or other of the cases con- 

 sidered must happen. 



When it is known beforehand that either A or B 

 must happen, and out of m + n times A has hap- 

 pened m times, and B n times, then (page 65.) it is 

 m -f 1 to n -f- 1 that A will happen the next time. 

 But suppose we have no reason, except what we gather 

 from the observed event, to know that A or B must 

 happen ; that is, suppose C or D, or E, &c. might have 

 happened: then the next event may be either A or 

 B, or a new species, of which it can be found that the 

 respective probabilities are proportional to m + 1, 

 w-j-1, and 1 ; so that though the odds remain w-fl 

 1 to n + 1 for A rather than B, yet it is now m + 1 

 to n + 2 for A against either B or the new event. 

 Thus, suppose a game at which one party or the other 

 must win, and suppose that out of 20 games A has 

 won 1 3 and B 7 : and this is all we know of the game 

 or of the players. Then, it is 134-1 to 7-fl> or 

 14 to 8, or 7 to 4, that A shall win the 21st game. 

 But suppose that it is possible to have a drawn game ; 

 then there is some chance that the 21st may be a 

 drawn game, though but a small one, as might be 

 inferred from such a thing never happening in 20 

 trials. The 21st game may be either A's or B's, or 

 drawn: of which the chances are as 15 + 1, 7+1, 

 and 1 ; or as 1 4, 8, and 1 . Consequently, though in 

 the preceding case it was 14 out of 22 in favour of A's 



