93 MONTMORT. 



SO that he had not really advanced beyond Pascal, although the 

 formula would be more convenient than the use of the Arith- 

 metical Triangle ; see Art. 23. The first formula for the case 

 of unequal skill was communicated to Montmort by John Ber- 

 noulli in a letter dated March 17th, 1710 ; see Montmort's page 295. 

 As we have already stated the formula was known to James 

 BernouUi; see Art. 113. The second formula for the Problem of 

 Points must be assigned to Montmort himself, for it now appears 

 before us for the first time. 



174. It will be interesting to make some comparison between 

 the two formulae given in Art. 172. 



It may be shewn that we have identically 



p' + rp''q-h \ ,^ ' J) <i +... + -—:::: rV 9. 



r^V^ if-^ + m (2> + ^)--^ + !!^!i±i) (^ + g)'-'-/ + 



r — 1 



m — 1 n 



This may be shewn by picking out the coefficients of the 

 various powers of ^ in the expression on the right-hand side, 

 making use of the relations presented by the identity 



(1 - j)-"-»'(l- 2)-= (1 -?)"'. 



Thus we see that \i 'p-\- c[ be equal to unity the two expres- 

 sions given in Art. 172 for ^'s chance are numerically equal. 



175. If however ^ + ^^ be not equal to unity the two expres- 

 sions given in Art. 172 for ^'s chance are not numerically equal. 

 If we suppose jy-^- q less than unity, we can give the following in- 

 terpretation to the formulae. Suppose that A 's chance of winning 

 in a single trial is jp, and i?'s chance is q, and that there is the 

 chance ^—jp — q that it is a drawn contest. 



Then the formula 



mi, w (??z + 1) „ \r —\ 



^ 1.2 -^ m— 1^1 — 1^ 



