DE MOIVRE. 159 



easily be found by our Metliod : thus if tbe number of Packs be ^, the 

 Probability that one Card or more of the same Suit and Name in every 

 one of the Packs may be in the same position, will be expressed as fol- 

 lows, 



1 1 1 



n'-" 2[n(n- l)]''' ' [3 {w (ji - 1) (w - 2)]^-^ 



1 



[4 [n (71-1) {n-2) (n-S)\ 



—, &c. 



k-2. 



Laplace demonstrates this result; see Theorie . . . des Prob. 

 page 224. 



290. Problems xxxvii. and xxxviii. relate to the game of 

 Bowls; see Arts. 177, 250. 



De Moivre says, page 120, 



Having given formerly the Solution of this Problem, proposed to me 

 by the Honourable Francis Rohartes, Esq;, in the FhilosopTiical Trans- 

 actions Number 329; I there said, by way of Corollary, that if the 

 proportion of Skill in the Gamesters were given, the Problem might 

 also be solved : since w^hich time M. de Monmort^ in the second Edition 

 of a Book by him published upon the Subject of Chance, has solved 

 this Problem as it is extended to the consideration of the Skill, and 

 to carry his Solution to a great number of Cases, giving also a Me- 

 thod whereby it might be carried farther: But altlio' his Solution is 

 good, as he has made a right use of the Doctrine of Combinations, 

 yet I think mine has a greater degree of Simplicity, it being deduced 

 from the original Principle whereby I have demonstrated the Doctrine 

 of Permutations and Combinations:... 



291. Problems xxxix. to XLil. form a connected set. Pro- 

 blem XXXIX. is as follows : 



To find the Expectation of A, when with a Die of any given num- 

 ber of Paces, he undertakes to fling any number of them in any given 

 number of Casts. 



Let j9 -f 1 be the number of faces on the die, n the number 

 of casts, /the number of faces which A undertakes to fling. Then 

 -4's expectation is 



