158 DE MOIVRE. 



pulled out, A shall give B a Guinea; and it were required to find what 

 consideration £ ought to give A to play on those Terms : the Answer 

 will be one Guinea, let the number of Cards be what it will. 



Altho' this be a Corollary from the preceding Solutions, yet it may 

 more easily be made out thus ; one of the Packs being the Rule where- 

 by to estimate the order of the Cards in the second, the Probability 



that the two first Cards are alike is — , the Probability that the two 



1^ 



52 



second are alike is also -^ , and therefore there being 52 such alike com- 



52 

 binations, it follows that the value of the whole is r-:: == 1. 



52 



It may be interesting to deduce this result from the formulae 

 already given. The chance that out of ti cards, 2^ specified cards 

 will be in their places, and all the rest out of their places will 

 be obtained by making q= n —p in the first formula of Art. 281. 

 The chance that cmy p cards will be in their places, and all the 

 rest out of their places will be obtained by multiplying the pre- 



ceding result by - — ^= . And since in this case B receives 



\n — p I p 



p guineas, we must multiply by p to obtain 5's advantage. Thus 

 we obtain 



p-l \ '2 [3 ' [^ 



n — p 



Denote this by <^ {p) ; then we are to shew that the sum of 

 the values of <^ {p) obtained by giving to p all values between 

 1 and n inclusive is unity. 



Let y^r (ti) denote the sum ; then it may be easily shewn that 



'f(n + l)-'f {71) = 0. 



Thus -yjr (n) is constant for all values of n ; and it = 1 when 

 71 = 1, so that -^ {71) is always = 1. 



289. The sixth Corollary to Problem xxxvi. is as follows : 

 If the number of Packs be given, the Probability that any given 

 number of Circumstances may happen iu any number of Packs, will 



