LAPLACE. 599 



Thus there is an advantage in undertaking to throw n heads 

 in succession beyond what there would be if the coin were per- 

 fectly symmetrical. 



Laplace shews how we may diminish the influence of the want 

 of symmetry in a coin. 



Let there be two coins A and B', let the chances of head 

 and tail in ^ be ^ and q respectively, and in B let them be p 

 and q respectively : and let us determine the probability that in 

 n throws the two coins shall always exhibit the same faces. 



The chance required is {pp + qqY. 



Suppose that 



1 +a 1-a 



, 1 + a' , 1-a' 



then {pp + qqf = ^. (1 + «« )" • 



But as we do not know to which faces the want of S3nTimetry 

 is favourable, the preceding expression might also be »^ (1 — aa')" 



by interchanging the forms of p and q or of p and q. Thus 

 the true value will be 



iji(l + aar+i(l-«a')''}, 



that is 



1 f n{n-l) , n{n-\)[n-^){n-Z) ,„ { 

 2-„|l+ j_2 aa + g aa +...^. 



It is obvious that this expression is nearer to — than that 



which was found for the probability of securing n heads in 7i 

 throws with a single coin. 



1033. Laplace gives again the result which we have noticed 

 in Art. 891. Suppose p to denote ^'s skill, and q to denote B's 

 skill ; let A have originally a counters and B have originally b 

 counters. Then A's chance of ruining B is 



