120 MONTMORT. 



203. The next letter is from Montmort to John Bernoulli ; it 

 occupies pages 803 — 307. Montmort makes brief observations on 

 the points to which John Bernouilli had drawn his attention ; he 

 suggests a problem on the Duration of Play for the consideration 

 of Nicolas Bernoulli. 



204. The next letter is from Nicolas Bernoulli to Montmort ; 

 it occupies pages 808 — 814. 



Nicolas Bernoulli first speaks of the game of Treize, and gives 

 a general formula for it ; but by accident he gave the formula in- 

 correctly, and afterwards corrected it w^hen Montmort drew his 

 attention to it ; see Montmort's pages 815, 328. 



We will here investigate the formula after the manner given by 

 Nicolas BernoulH for the simple case already considered in Art. 161. 



Suppose there are n cards divided into p sets. Denote the 

 cards of a set by a,h,c,... in order. 

 The whole number of cases is \n. 

 The number of ways in which a can stand first is p \n — \ . 



The number of ways in which h can stand second without a 

 standing first is p \n — l — p\ n 



The number of ways in which c can stand third without a 

 standing first or h second is p \n — \ — 2p^ |^ — 2 + p^ | n — 3 . 

 And so on. 



Hence the chance of winning by the first card is - ; the chance 



n 



of winning: by the second card is -, ^ ,. ; the chance of Avin- 



° -^ n 7i{n— 1) 



ning by the third card is — , ^ .,. H 7 =^-7 -^ ; and so on. 



° '' n n{n—l) n{n— 1) [ii — z) 



Hence the chance of winning by one or other of the first m 

 cards is 



mjj m (m — 1) p^ m (m — 1) {m — 2) p^ 



"^"' O n {n - 1) "^ 1.2.3 7i (n - 1) (w -2) "" *'* 



And the entire chance of winning is found by putting 



m = - , so that it is 

 P 



